Chapter 3 of 8

38%

NDA Mathematics★ ~15% of the Mathematics paper — exactly 18 of 120 Q (45 of 300 marks), NDA's third-highest-weight Maths topic after Algebra (20%) and Trigonometry (18%)

Analytical Geometry — Lines, Circles, Conics & Three Dimensions

Analytical Geometry is NDA Maths' third-highest-weight topic — about 18 of 120 questions, worth roughly 45 marks — and it is really two exams stitched into one. Two-dimensional coordinate geometry (distance, lines, circles, conics) is built on algebra you already know; three-dimensional geometry (direction cosines, lines, planes) extends the same handful of ideas by one extra coordinate. No calculator, no proofs, just correct substitution into the right formula under a 1/3rd-mark penalty.

⏱ 48 min readmedium-hard difficulty✓ Free · exam-pattern aligned

Timed practiceMCQs from this chapter with instant scoring

FlashcardsFormulas & traps on spaced repetition

NDA PYQsSolved previous-year questions

By the end of this chapter you'll be able to…

1Apply the distance and section formulas (internal/external, midpoint, centroid) to solve 2D coordinate-geometry problems, and use the area-of-a-triangle formula to test collinearity
2Derive and apply the six standard forms of a straight line's equation, compute the angle between two lines and the distance of a point from a line, and use the family-of-lines idea and the concurrency determinant
3Analyse a pair of straight lines given by a homogeneous second-degree equation (angle between them, perpendicularity, coincidence), and extract a circle's centre and radius from its standard or general equation including tangency conditions
4Recall the standard equations, eccentricity, focus, directrix and latus rectum of the parabola, ellipse and hyperbola at NDA's application level (not JEE's chord/tangent/normal depth)
5Extend distance, section formula and direction concepts to three dimensions using direction cosines and direction ratios, and write the symmetric form of a line in space
6Apply the equation of a plane in various forms to find the angle between two planes, the angle between a line and a plane, and the distance of a point from a plane

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Why this chapter matters in NDA

Analytical Geometry carries 15% of NDA Mathematics — 18 of the paper's 120 questions and exactly 45 of 300 marks, the third-heaviest topic after Algebra and Trigonometry. It is really two chapters fused into one: two-dimensional coordinate geometry (distance, lines, circles, and the three conics at a shallow standard-equation level) built directly on algebra already known, and three-dimensional geometry (direction cosines, lines, planes) that extends the same handful of ideas by one coordinate. Neither half asks for insight — every question hands over numbers and expects the correct named formula applied without a sign slip. With wrong answers costing 2.5/3 ≈ 0.8333 marks, the chapter punishes exactly two habits: reusing a 2D formula unchanged in 3D (dropping the z-term), and confusing near-identical formula pairs (internal vs external section formula, two-line angle vs pair-of-lines angle, cosine for plane-plane angle vs sine for line-plane angle). Master the formula list and the discipline of matching the right one to the right question, and this becomes some of the fastest, most reliable marks in the paper.

Before you start — revise these

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Algebra (NDA Mathematics)

Solving linear and quadratic equations is used constantly — finding intersections of lines, applying tangency conditions to circles, and simplifying eccentricity expressions all reduce to algebra you should already have automatic.

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Trigonometry (NDA Mathematics)

Slope is literally tanθ, and the angle-between-two-lines/planes formulas are trigonometric ratios in disguise — comfort with standard angle values speeds up recognising clean answers like 30°, 45°, 60°.

Analytical Geometry — NDA Mathematics

Analytical geometry at NDA level never asks you to discover anything — every question hands you coordinates, a line, a circle, a conic, or a plane, and asks you to plug them into a named formula correctly and fast. The subject has two halves that share one grammar: two-dimensional coordinate geometry (points, lines, circles, and the three conics at their shallowest, standard-equation level) and three-dimensional geometry (the same distance/section-formula ideas extended by a $z$ -coordinate, plus lines and planes in space). Master the dozen or so core formulas, know exactly which one a question is pointing at, and this chapter becomes fast, reliable marks.

1. What NDA actually asks

Analytical Geometry carries weightPct 15 of NDA Mathematics — third only to Algebra (20%) and Trigonometry (18%), and ahead of both calculus topics. Against 120 questions at 2.5 marks each, that is exactly 18 questions worth exactly 45 marks in a typical paper, spread across seven recognisable families:

Coordinate basics — distance formula, section formula (internal/external), area of a triangle, and collinearity.
Straight lines — every standard form of the equation, slope, angle between two lines, distance of a point from a line, family of lines, and concurrency.
Pair of straight lines — the homogeneous second-degree equation $a x^{2} + 2 h x y + b y^{2} = 0$ and the angle it represents.
Circles — standard and general equations, centre/radius extraction, and tangency conditions.
Conic sections — parabola, ellipse and hyperbola at their standard-equation level: eccentricity, focus, directrix, latus rectum — never the deep chord/tangent/normal machinery JEE tests.
Three-dimensional coordinate geometry — distance between two points, section formula, direction cosines and direction ratios.
Lines and planes in space — the symmetric form of a line, the equation of a plane in various forms, angle between two lines/planes, and distance of a point from a plane.

Because a wrong answer costs $2.5/3 \approx 0.8333$ marks, this chapter punishes a particular kind of carelessness: reusing a familiar 2D formula unchanged in a 3D question (forgetting the $z$ -term), or mixing up which sign convention a formula uses (internal vs external section formula, $(- g, - f)$ vs $(g, f)$ for a circle's centre). The mathematics itself is rarely hard — the discipline of matching the right formula to the right question is what's actually being tested.

2. Coordinates, distance & section formula (2D)

For two points $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2})$ , the distance formula is Pythagoras applied to the horizontal and vertical legs of the segment $P Q$ :

$P Q = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

Section formula. The point $R$ dividing $P Q$ in the ratio $m : n$ internally (between $P$ and $Q$ ) is

$R = (\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$

and externally (on the extension of $P Q$ , beyond $Q$ ) is

$R = (\frac{m x _{2} - n x _{1}}{m - n}, \frac{m y _{2} - n y _{1}}{m - n})$

Only the sign changes between the two formulas — but that single sign swap moves $R$ from inside the segment to outside it, which makes internal/external confusion the single most common error in this section. Midpoint (the special case $m = n$ ): $(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$ . Centroid of a triangle with vertices $(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})$ : $(\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})$ — the average of all three vertices, a very frequent one-line NDA question.

3. Area of a triangle & collinearity

For vertices $(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})$ :

$Δ = \frac{1}{2} x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})$

Three points are collinear exactly when this vanishes — $Δ = 0$ is NDA's favourite way to disguise a "prove three points are collinear" question as an MCQ: instead of a proof, it just asks you to compute the area and check that it's zero.

4. The straight line — slope and forms of the equation

Slope of the line joining $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ : $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = tan θ$ , where $θ$ is the angle the line makes with the positive $x$ -axis.

Form	Equation	When to use it
Slope-intercept	$y = m x + c$	slope and $y$ -intercept known
Point-slope	$y - y_{1} = m (x - x_{1})$	slope and one point known
Two-point	$\frac{y - y _{1}}{x - x _{1}} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$	two points known
Intercept	$\frac{x}{a} + \frac{y}{b} = 1$	$x$ -intercept $a$ , $y$ -intercept $b$ known
Normal	$x cos α + y sin α = p$	perpendicular distance $p$ from origin and its inclination $α$ known
General	$a x + b y + c = 0$	slope $= - a / b$ ; the form everything else reduces to

All six describe the same object — a straight line — from different given data; NDA questions are really testing whether you can pick the right one and substitute cleanly.

5. Angle between lines, distance from a line, concurrency & families

Angle between two lines with slopes $m_{1}, m_{2}$ :

$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$

Parallel: $m_{1} = m_{2}$ . Perpendicular: $m_{1} m_{2} = - 1$ . This formula gives the acute angle directly (via the absolute value); dropping the absolute value and reading off the raw arctangent gives the obtuse companion angle instead — both are genuine angles between the lines, but only one is usually asked for.

Distance of a point $(x_{1}, y_{1})$ from a line $a x + b y + c = 0$ :

$d = \frac{∣ a x _{1} + b y _{1} + c ∣}{a ^{2} + b ^{2}}$

Distance between two parallel lines $a x + b y + c_{1} = 0$ and $a x + b y + c_{2} = 0$ : $d = \frac{∣ c _{1} - c _{2} ∣}{a ^{2} + b ^{2}}$ .

Family of lines. Every line through the intersection of $L_{1} \equiv a_{1} x + b_{1} y + c_{1} = 0$ and $L_{2} \equiv a_{2} x + b_{2} y + c_{2} = 0$ (except $L_{2}$ itself) can be written as $L_{1} + λ L_{2} = 0$ for some real $λ$ — a shortcut that avoids solving for the intersection point explicitly when a further condition (e.g. "passes through a third point") pins down $λ$ .

Concurrency of three lines $a_{1} x + b_{1} y + c_{1} = 0$ , $a_{2} x + b_{2} y + c_{2} = 0$ , $a_{3} x + b_{3} y + c_{3} = 0$ : they meet at a single point exactly when

$a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} = 0$

In practice, the faster route for an MCQ is usually to find the intersection of any two lines directly and substitute into the third — the determinant is a good cross-check when time allows.

6. Pair of straight lines

A homogeneous second-degree equation $a x^{2} + 2 h x y + b y^{2} = 0$ always represents two straight lines through the origin (real, coincident, or imaginary depending on the discriminant). Factor it as a product of two linear factors to read off the individual lines, or use:

$tan θ = \frac{2 h ^{2} - ab}{a + b}$

for the angle between them. Lines coincide when $h^{2} = ab$ ; lines are perpendicular when $a + b = 0$ (compare this to the two-line perpendicularity condition $m_{1} m_{2} = - 1$ — same idea, different formula, and NDA distractors love swapping one for the other). A general second-degree equation $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ represents a pair of straight lines (not necessarily through the origin) exactly when $ab c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$ , equivalently $a h g h b f g f c = 0$ — but NDA tests the origin-through case (no $g, f, c$ ) far more often.

7. Circles

Standard form, centre $(h, k)$ and radius $r$ : $(x - h)^{2} + (y - k)^{2} = r^{2}$ .

General form: $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ . Completing the square gives $(x + g)^{2} + (y + f)^{2} = g^{2} + f^{2} - c$ , so

$centre = (- g, - f), radius = g^{2} + f^{2} - c$

The minus signs in the centre are easy to drop under time pressure — always negate both halves of the coefficients of $x$ and $y$ (divided by 2), never just copy $g$ and $f$ directly.

Tangency. A line touches a circle exactly when the perpendicular distance from the centre to the line equals the radius — reuse the point-to-line distance formula from Section 5 with the centre as the point. For the line $y = m x + c$ and the circle $x^{2} + y^{2} = a^{2}$ , this condition simplifies to $c^{2} = a^{2} (1 + m^{2})$ . The tangent to $x^{2} + y^{2} = a^{2}$ at a point $(x_{1}, y_{1})$ on the circle is $x x_{1} + y y_{1} = a^{2}$ (replace $x^{2} \to x x_{1}$ and $y^{2} \to y y_{1}$ — the standard " $T = 0$ " trick).

8. Conic sections — parabola, ellipse, hyperbola

NDA tests these at the standard-equation level only — no chords, tangents, or normals beyond what's listed here.

Parabola $y^{2} = 4 a x$ (opens rightward, vertex at origin): focus $(a, 0)$ , directrix $x = - a$ , latus rectum length $4 a$ . The three sibling forms $y^{2} = - 4 a x$ , $x^{2} = 4 a y$ , $x^{2} = - 4 a y$ open left, up, and down respectively — the axis of symmetry is always the squared variable's axis, and the sign fixes the direction.

Ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ with $a > b$ (major axis along $x$ ): foci $(\pm a e, 0)$ where $e = \frac{c}{a} = 1 - \frac{b ^{2}}{a ^{2}}$ and $c^{2} = a^{2} - b^{2}$ ; directrices $x = \pm a / e$ ; latus rectum length $\frac{2 b ^{2}}{a}$ . If instead $b > a$ , the major axis is along $y$ and every formula swaps roles — always check which denominator is larger before assuming $a$ is the semi-major axis.

Hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ : foci $(\pm a e, 0)$ where $e = \frac{c}{a} = 1 + \frac{b ^{2}}{a ^{2}}$ and $c^{2} = a^{2} + b^{2}$ (note the plus, not minus, unlike the ellipse); directrices $x = \pm a / e$ ; latus rectum $\frac{2 b ^{2}}{a}$ ; eccentricity is always $> 1$ (versus $< 1$ for an ellipse). Asymptotes: $y = \pm \frac{b}{a} x$ .

9. Coordinates, distance & section formula (3D)

Everything in Section 2 extends by carrying a third coordinate through unchanged in structure — the trap is forgetting to actually include it.

Distance between $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$ :

$P Q = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$

Section formula (internal), ratio $m : n$ : $(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n}, \frac{m z _{2} + n z _{1}}{m + n})$ — external is the same sign-swap as in 2D. Centroid of a triangle with vertices $(x_{i}, y_{i}, z_{i})$ : average each coordinate, exactly as in 2D.

Direction cosines $(l, m, n)$ of a line are the cosines of the angles it makes with the $x$ , $y$ , $z$ axes, and always satisfy $l^{2} + m^{2} + n^{2} = 1$ . Direction ratios $(a, b, c)$ are any triple proportional to $(l, m, n)$ — components of a vector along the line qualify directly, with no normalisation required. Converting ratios to cosines:

$l = \frac{a}{a ^{2} + b ^{2} + c ^{2}}, m = \frac{b}{a ^{2} + b ^{2} + c ^{2}}, n = \frac{c}{a ^{2} + b ^{2} + c ^{2}}$

The direction ratios of the line joining $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ are simply $(x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1})$ — the same subtraction that feeds the distance formula, reused.

10. The straight line in three dimensions

A line in space needs a point and a direction — there is no single "slope" the way there is in 2D. Through a point $(x_{1}, y_{1}, z_{1})$ with direction ratios $(a, b, c)$ , the symmetric (Cartesian) form is

$\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = \frac{z - z _{1}}{c}$

Through two points $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ , replace the direction ratios with $(x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1})$ .

Angle between two lines with direction ratios $(a_{1}, b_{1}, c_{1})$ and $(a_{2}, b_{2}, c_{2})$ :

$cos θ = \frac{a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2}}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$

Perpendicular: $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$ . Parallel: $\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}$ .

11. The plane

General form: $a x + b y + cz + d = 0$ , with $(a, b, c)$ as the normal vector to the plane — the single most useful fact in this section, since every other plane formula is built from comparing normals.

Point + normal form: through $(x_{1}, y_{1}, z_{1})$ with normal $(a, b, c)$ : $a (x - x_{1}) + b (y - y_{1}) + c (z - z_{1}) = 0$ . Intercept form: $\frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 1$ , with intercepts $p, q, r$ on the three axes.

Angle between two planes with normals $(a_{1}, b_{1}, c_{1})$ and $(a_{2}, b_{2}, c_{2})$ — identical in form to the angle between two lines, just applied to the normals instead:

$cos θ = \frac{a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2}}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$

Planes are parallel when their normals are proportional, perpendicular when the normals' dot product is zero.

Distance of a point $(x_{1}, y_{1}, z_{1})$ from a plane $a x + b y + cz + d = 0$ — the direct 3D analogue of the point-to-line distance formula:

$d = \frac{∣ a x _{1} + b y _{1} + c z _{1} + d ∣}{a ^{2} + b ^{2} + c ^{2}}$

One subtlety worth flagging: the angle between a line (direction ratios $a, b, c$ ) and a plane (normal $A, B, C$ ) uses sine, not cosine — $sin θ = \frac{∣ a A + b B + c C ∣}{a ^{2} + b ^{2} + c ^{2} A ^{2} + B ^{2} + C ^{2}}$ — because the angle is conventionally measured between the line and the plane itself, which is the complement of the angle between the line and the plane's normal.

12. Solved PYQ-style examples

Q1. Find the distance between $(- 2, 3)$ and $(4, - 5)$ . Solution. $(4 - (- 2))^{2} + (- 5 - 3)^{2} = 36 + 64 = 100 = 10$ .

Q2. Find $k$ if $(1, 2)$ , $(k, 4)$ , $(5, 6)$ are collinear. Solution. Area $= 0$ : $1 (4 - 6) + k (6 - 2) + 5 (2 - 4) = 0 \Rightarrow - 2 + 4 k - 10 = 0 \Rightarrow k = 3$ .

Q3. Find the equation of the line through the origin parallel to $3 x - 4 y + 7 = 0$ . Solution. A parallel line has the same $x, y$ coefficients: $3 x - 4 y + c = 0$ . Through $(0, 0)$ : $c = 0$ , so $3 x - 4 y = 0$ .

Q4. Find $tan θ$ for the pair of lines $x^{2} - 3 x y + 2 y^{2} = 0$ . Solution. Factor: $(x - y) (x - 2 y) = 0$ , giving slopes $m_{1} = 1, m_{2} = \frac{1}{2}$ . Directly: $tan θ = \frac{1 - 1/2}{1 + 1/2} = \frac{1/2}{3/2} = \frac{1}{3}$ . (Cross-check via the formula: $a = 1, b = 2, h = - 3/2$ , $tan θ = \frac{2 9/4 - 2}{3} = \frac{2 1/4}{3} = \frac{1}{3}$ — matches.)

Q5. Find the centre and radius of $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ . Solution. $g = - 2, f = 3, c = - 12$ . Centre $= (2, - 3)$ , radius $= 4 + 9 + 12 = 25 = 5$ .

Q6. Find the eccentricity of $\frac{x ^{2}}{16} - \frac{y ^{2}}{9} = 1$ . Solution. $a^{2} = 16, b^{2} = 9$ , $c^{2} = a^{2} + b^{2} = 25 \Rightarrow c = 5$ , $e = c / a = 5/4$ .

Q7. Find the direction cosines of the line joining $(1, 2, - 1)$ and $(3, - 1, 2)$ . Solution. Direction ratios: $(2, - 3, 3)$ . Magnitude $= 4 + 9 + 9 = 22$ . Direction cosines: $(\frac{2}{22}, \frac{- 3}{22}, \frac{3}{22})$ .

Q8. Find the distance of $(1, 2, 3)$ from the plane $2 x - y + 2 z + 6 = 0$ . Solution. $∣2 (1) - 2 + 2 (3) + 6∣ = ∣2 - 2 + 6 + 6∣ = 12$ . Denominator $= 4 + 1 + 4 = 3$ . Distance $= 12/3 = 4$ .

13. Common traps and exam protocol

Internal vs external section formula. The sign flip between $\frac{m x _{2} + n x _{1}}{m + n}$ and $\frac{m x _{2} - n x _{1}}{m - n}$ moves the point from inside the segment to outside it entirely — read the word "internally"/"externally" before picking the formula, never assume internal by default.
Circle centre sign. From $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , the centre is $(- g, - f)$ , not $(g, f)$ . Always negate both halves of the linear coefficients.
Two-line vs pair-of-lines angle formulas. $tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$ (two separate given lines) and $tan θ = \frac{2 h ^{2} - ab}{a + b}$ (one homogeneous equation representing both lines at once) are not interchangeable — match the formula to how the lines are actually given.
Ellipse/hyperbola axis mix-up. Don't assume $a^{2}$ is always the larger denominator — check which one actually is, since $e = c / a$ flips into a very different (and for an ellipse, potentially invalid, $> 1$ ) number if $a$ and $b$ are swapped.
Direction ratios treated as direction cosines. Direction ratios need not satisfy $l^{2} + m^{2} + n^{2} = 1$ ; only the magnitude-normalised version does. Divide by $a^{2} + b^{2} + c^{2}$ before calling something a direction cosine.
Dropping the $z$ -term in 3D. The single most common transition error — applying the 2D distance or section formula out of habit and forgetting the third coordinate exists at all.
Line-plane angle uses sine, not cosine. Unlike the line-line and plane-plane angle formulas (both cosine), the angle between a line and a plane is found via $sin θ$ , since it's measured from the plane itself, not from the plane's normal.
Forgetting the absolute value and the square-root denominator in the point-to-line and point-to-plane distance formulas — both a sign slip and a missing $\cdot$ are extremely common under time pressure.

Protocol: spend the first pass memorising the six line forms and the three distance formulas (point-to-line, point-to-plane, and between two points in both 2D and 3D) until substitution is automatic. Then drill the circle centre/radius extraction and the conic standard-form table (focus, directrix, latus rectum, eccentricity) as pure recall. Finish with 10–15 mixed 3D problems specifically, since that's the newest material for most students and the place where habitual 2D shortcuts cause the most avoidable errors.

Key formulas & results

Everything to memorise for the exam hall, in one card. Screenshot this for revision.

Distance formula (2D)

PQ = √[(x2−x1)² + (y2−y1)²]

Pythagoras on the horizontal and vertical legs of the segment — the seed formula for almost everything else in the chapter.

Section formula (2D)

Internal, ratio m:n: ((mx2+nx1)/(m+n), (my2+ny1)/(m+n)) · External: ((mx2−nx1)/(m−n), (my2−ny1)/(m−n)) · Midpoint: ((x1+x2)/2, (y1+y2)/2) · Centroid: ((x1+x2+x3)/3, (y1+y2+y3)/3)

Only a sign changes between internal and external — but it moves the point from inside the segment to outside it entirely. Read the wording before choosing.

Area of a triangle / collinearity

Δ = ½ |x1(y2−y3) + x2(y3−y1) + x3(y1−y2)|

Three points are collinear exactly when Δ = 0 — NDA's favourite disguised way to ask a collinearity question as an MCQ.

Forms of a straight line

y=mx+c (slope-intercept) · y−y1=m(x−x1) (point-slope) · (y−y1)/(x−x1)=(y2−y1)/(x2−x1) (two-point) · x/a+y/b=1 (intercept) · xcosα+ysinα=p (normal) · ax+by+c=0 (general, slope=−a/b)

All six describe the same object from different given data — the skill being tested is picking the right one and substituting cleanly.

Angle between two lines / distance from a line

tanθ = |(m1−m2)/(1+m1m2)| · Parallel: m1=m2 · Perpendicular: m1m2=−1 · Distance of (x1,y1) from ax+by+c=0: |ax1+by1+c|/√(a²+b²)

Dropping the absolute value in the angle formula gives the obtuse companion angle instead of the acute one usually asked for.

Family of lines & concurrency

Family through L1∩L2: L1 + λL2 = 0 · Three lines concurrent iff |a1 b1 c1; a2 b2 c2; a3 b3 c3| = 0

Faster in practice: find the intersection of any two lines directly and substitute into the third; use the determinant as a cross-check.

Pair of straight lines through the origin

ax² + 2hxy + by² = 0 represents two lines through the origin · tanθ = 2√(h²−ab)/(a+b) · Coincident: h²=ab · Perpendicular: a+b=0

This angle formula is NOT the same as the two-line tanθ formula (m1,m2) — match the formula to how the lines are given (one combined equation vs two separate ones).

Circle — standard & general form

Standard: (x−h)²+(y−k)²=r², centre (h,k), radius r · General: x²+y²+2gx+2fy+c=0, centre (−g,−f), radius √(g²+f²−c)

Always negate both g and f to get the centre — copying (g,f) directly instead of (−g,−f) is the most common circle error.

Tangent condition & tangent equation

Line y=mx+c tangent to x²+y²=a² iff c²=a²(1+m²) · Tangent to x²+y²=a² at (x1,y1) on the circle: xx1+yy1=a²

A line touches a circle exactly when its perpendicular distance from the centre equals the radius — reuse the point-to-line distance formula with the centre as the point.

Parabola (standard form)

y²=4ax: vertex (0,0), focus (a,0), directrix x=−a, latus rectum 4a

y²=−4ax, x²=4ay, x²=−4ay open left/up/down respectively — the squared variable's axis is always the axis of symmetry.

Ellipse (standard form, a>b)

x²/a² + y²/b² = 1, foci (±ae,0), c²=a²−b², e=c/a=√(1−b²/a²) <1, directrices x=±a/e, latus rectum 2b²/a

If b>a instead, the major axis is along y and every formula swaps roles — check which denominator is actually larger first.

Hyperbola (standard form)

x²/a² − y²/b² = 1, foci (±ae,0), c²=a²+b² (PLUS, unlike the ellipse), e=c/a=√(1+b²/a²) >1, directrices x=±a/e, latus rectum 2b²/a, asymptotes y=±(b/a)x

Eccentricity is always greater than 1 for a hyperbola, always less than 1 for an ellipse — a quick sanity check on any computed value.

Distance & section formula (3D)

PQ = √[(x2−x1)²+(y2−y1)²+(z2−z1)²] · Internal section (ratio m:n): ((mx2+nx1)/(m+n), (my2+ny1)/(m+n), (mz2+nz1)/(m+n))

Identical in structure to the 2D versions with a z-term appended — the trap is forgetting to actually include it, not the formula itself.

Direction cosines & direction ratios

Direction cosines (l,m,n): l²+m²+n²=1 · Converting ratios (a,b,c) to cosines: l=a/√(a²+b²+c²), similarly for m,n · DRs of the line joining (x1,y1,z1),(x2,y2,z2): (x2−x1, y2−y1, z2−z1)

Direction ratios need NOT satisfy l²+m²+n²=1 — only the magnitude-normalised direction cosines do. Never report DRs as if they were DCs.

Line in 3D & angle between two lines

Symmetric form through (x1,y1,z1) with DRs (a,b,c): (x−x1)/a=(y−y1)/b=(z−z1)/c · cosθ = (a1a2+b1b2+c1c2)/(√Σa1²·√Σa2²) · Perpendicular: a1a2+b1b2+c1c2=0 · Parallel: a1/a2=b1/b2=c1/c2

There is no single 'slope' in 3D — a line needs a point plus a direction, always.

Plane — equation forms & angle between planes

General: ax+by+cz+d=0, normal (a,b,c) · Point+normal: a(x−x1)+b(y−y1)+c(z−z1)=0 · Intercept: x/p+y/q+z/r=1 · Angle between planes (normals n1,n2): cosθ=(n1·n2)/(|n1||n2|)

Identical form to the line-line angle formula, just applied to the two normal vectors instead of the two direction-ratio vectors.

Distance of a point from a plane; line-plane angle

Distance of (x1,y1,z1) from ax+by+cz+d=0: |ax1+by1+cz1+d|/√(a²+b²+c²) · Angle between a line (DRs a,b,c) and a plane (normal A,B,C): sinθ=|aA+bB+cC|/(√Σa²·√ΣA²)

Line-plane angle uses SINE, not cosine — it's measured from the plane itself, the complement of the angle between the line and the plane's normal.

NDA test series

Ready to be tested on this?

Analytical Geometry — Lines, Circles, Conics & Three Dimensions shows up in every NDA paper — sit a timed one and find out where you actually stand.

16–20Qs in paper

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Traps NDA sets — and how to dodge them

These are the exact option-traps and misreads that cost marks under negative marking.

WATCH OUT

✗ Confusing the internal and external section formula

✓ The only difference is a sign: (mx2+nx1)/(m+n) for internal, (mx2−nx1)/(m−n) for external. Read whether the point divides the segment 'internally' or 'externally' before picking the formula — the two give very different points.

WATCH OUT

✗ Taking a circle's centre as (g,f) instead of (−g,−f) from x²+y²+2gx+2fy+c=0

✓ Always negate BOTH halved coefficients. Write '(−g,−f)' as a fixed phrase before touching the numbers, so the negation isn't a separate step you can forget.

WATCH OUT

✗ Using the two-line angle formula tanθ=(m1−m2)/(1+m1m2) and the pair-of-lines formula tanθ=2√(h²−ab)/(a+b) interchangeably

✓ They apply to different situations — two separately given lines (slopes m1,m2) versus one homogeneous second-degree equation ax²+2hxy+by²=0 representing both lines at once. Identify which form the question actually gives before choosing.

WATCH OUT

✗ Assuming a² is always the larger denominator in an ellipse or hyperbola equation

✓ Check which denominator is actually larger before computing e=c/a — for an ellipse with the major axis along y (b>a), every formula swaps roles, and blindly using the smaller value as a² gives an eccentricity above 1, which is impossible for an ellipse.

WATCH OUT

✗ Reporting direction ratios as if they were direction cosines

✓ Direction cosines must satisfy l²+m²+n²=1; direction ratios don't need to. Always divide each ratio by √(a²+b²+c²) before calling the result a direction cosine.

WATCH OUT

✗ Applying a 2D distance or section formula in a 3D question without adding the z-term

✓ This is the single most common 3D-transition error. Before writing any 3D formula, consciously write out all three coordinate differences (Δx, Δy, Δz) — don't let 2D habit silently drop the third.

WATCH OUT

✗ Using cosine for the angle between a line and a plane, the same as for line-line or plane-plane angles

✓ Line-plane angle uses SINE: sinθ=|aA+bB+cC|/(√Σa²·√ΣA²), because the angle is measured from the plane itself, which is the complement of the angle to the plane's normal. Line-line and plane-plane angles both use cosine — only line-plane is the odd one out.

WATCH OUT

✗ Dropping the absolute value or the square-root denominator in the point-to-line or point-to-plane distance formula

✓ Distance is always non-negative — the |·| in the numerator and the √(a²+b²[+c²]) in the denominator are both mandatory, non-optional parts of the formula, not simplifications you can skip when the sign already looks positive.

Exam-pattern practice

PYQ-style questions with full solutions. Work through them as a readiness check — mark yourself honestly and get your gap report at the end.

Readiness check

Are you exam-ready for Analytical Geometry — Lines, Circles, Conics & Three Dimensions?

16 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

16 questions~11 min worth ~5 marks in NDA exams

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Distance (2D): √[(x2−x1)²+(y2−y1)²]. Distance (3D): add a (z2−z1)² term — same structure, one more coordinate.
•Section formula: internal uses (mx2+nx1)/(m+n); external flips one sign to (mx2−nx1)/(m−n) — read the wording before choosing.
•Area of a triangle Δ=0 ⇔ collinear — the standard disguised-collinearity MCQ.
•Line forms: slope-intercept, point-slope, two-point, intercept, normal, general (ax+by+c=0, slope=−a/b) — pick based on what's given.
•tanθ=|(m1−m2)/(1+m1m2)| for two given lines; 2√(h²−ab)/(a+b) for a homogeneous pair ax²+2hxy+by²=0 — NOT interchangeable.
•Distance of a point from a line: |ax1+by1+c|/√(a²+b²). Same structure in 3D for a plane, with a z-term and c-term added.
•Circle general form x²+y²+2gx+2fy+c=0: centre (−g,−f), radius √(g²+f²−c) — always negate g and f.
•Parabola y²=4ax: focus (a,0), directrix x=−a, latus rectum 4a. Ellipse/hyperbola: e=c/a, with c²=a²−b² (ellipse, e<1) or c²=a²+b² (hyperbola, e>1) — the sign is the only difference.
•Direction ratios ≠ direction cosines until divided by √(a²+b²+c²); direction cosines always satisfy l²+m²+n²=1.
•Line in 3D has no single slope — needs a point plus direction ratios: (x−x1)/a=(y−y1)/b=(z−z1)/c.
•Plane ax+by+cz+d=0 has normal (a,b,c) — angle between two planes = angle between their normals (cosine formula, same shape as line-line angle).
•Line-plane angle is the ONE exception that uses sine instead of cosine, since it's measured from the plane, not from its normal.
•Distance of a point from a plane: |ax1+by1+cz1+d|/√(a²+b²+c²) — the direct 3D sibling of the point-to-line formula.

NDA question blueprint

How this topic is asked, tier by tier — so you can prep to the pattern.

Typical weightage: 45 of 300 Mathematics marks (18 questions × 2.5 marks, no partial credit)

Question style	Marks each	Typical count	What it tests
Coordinate basics — distance, section formula, area/collinearity	2.5	~3	Distance and section formula application, area-of-a-triangle collinearity check
Straight lines — forms, angle, distance, concurrency, pair of lines	2.5	~6	Equation of a line in various forms, angle between two lines, distance of a point from a line, concurrency, family of lines, pair-of-straight-lines angle
Circles	2.5	~3	Centre/radius from standard or general form, tangency conditions, tangent equation
Conic sections — parabola, ellipse, hyperbola	2.5	~3	Standard-equation recall, focus/directrix, latus rectum, eccentricity
Three-dimensional geometry — points, lines, planes	2.5	~3	3D distance/section formula, direction cosines/ratios, symmetric line form, plane equation, angle between lines/planes, distance from a plane

Prep strategy

Week 1: drill the 2D core — distance, section formula, area/collinearity, and all six line forms — until substitution is automatic; this feeds directly into circles and pair-of-lines questions later.
Week 2: master circles (centre/radius extraction, tangency) and the conic standard-form table (focus, directrix, latus rectum, eccentricity for all three conics) as pure recall, then move to 3D — distance, section formula, and direction cosines/ratios, consciously writing out the z-term every time.
In the final week, drill lines and planes in 3D (symmetric form, angle between lines/planes, distance from a plane) specifically, since this is the newest material for most students, and do 10-15 mixed problems switching rapidly between 2D and 3D to build the habit of matching the right formula to the right dimension.

Exam-hall strategy

Battle-tested tips from mentors and toppers for this topic under the sectional clock.

Before solving any line-angle or pair-of-lines question, identify whether the lines are given as two separate equations (use m1,m2) or as one homogeneous second-degree equation (use the h²−ab formula) — the wrong formula choice is the most avoidable error in this chapter.
For circle questions, write '(−g,−f)' as a fixed habit before substituting numbers — never let the negation become an afterthought.
Before any 3D question, consciously write out Δx, Δy, AND Δz — the single biggest speed-vs-accuracy trade-off in this chapter is 2D habit silently dropping the third coordinate.
For eccentricity questions, check which denominator (a² or b²) is actually larger before computing c²=a²−b² — an ellipse's eccentricity must come out less than 1; if it doesn't, the axes were swapped.
Remember the ONE sine exception: line-plane angle uses sinθ, while line-line and plane-plane angles both use cosθ — say this rule out loud before starting any angle-with-a-plane question.
No calculator is allowed — recognise clean angle values (30°, 45°, 60°, 90°) from standard tanθ/cosθ ratios (1/√3, 1, √3, undefined) instantly rather than computing an arctangent by hand.

Beyond the exam

Where this skill shows up in the job you're competing for — and in life.

Artillery trajectories and ballistics

A projectile fired under gravity traces a parabola — the same y²=4ax family this chapter teaches, just rotated and scaled — and understanding focus/directrix geometry underlies range-and-elevation calculations central to gunnery, directly relevant to NDA's Artillery and Infantry postings.

Satellite and planetary orbits

Every stable orbit — a satellite around Earth, a planet around the Sun — is an ellipse with the central body at one focus (Kepler's first law), making eccentricity a direct measure of how circular or elongated an orbit is; radar range circles and sonar detection zones use the plain circle equation the same way.

Navigation, surveying and GPS positioning

3D coordinate geometry — distance formula, direction cosines, and the equation of a plane — underlies GPS trilateration (fixing a position from distances to multiple satellites) and land/terrain surveying, where a levelled ground surface is modelled as a plane and bearings are direction ratios.

Structural and civil engineering

Analysing a roof truss, bridge deck, or bunker roofline in three dimensions reduces to writing planes and lines in space and computing angles between them — directly useful to the Corps of Engineers, and the same skill set that turns a blueprint's coordinates into buildable angles and distances.

Where else this topic is tested

Prepare once, score in every exam that asks it.

CDS (Combined Defence Services) Elementary MathematicsVery high — near-identical syllabus, weight, and question style

AFCAT (technical/numerical sections)Medium — lighter treatment, mostly 2D lines and circles, minimal 3D

JEE Main (Coordinate Geometry & 3D Geometry)Conceptual overlap — same formulas at much higher computational and conceptual depth (chords, tangents, normals)

CUET MathematicsHigh — same NCERT-level content and MCQ question style

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Questions aspirants ask

Pulled from the Q&A community and mentor sessions.

Much shallower. NDA sticks to the standard-equation level — recognising y²=4ax, x²/a²+y²/b²=1, x²/a²−y²/b²=1, and recalling focus, directrix, eccentricity and latus rectum for each. There are no chord-of-contact, tangent/normal-at-a-point, or director-circle questions the way JEE tests conics — just clean recall and substitution.

Yes, disproportionately so. 3D questions at NDA level are almost pure formula substitution with very little ambiguity about which formula to use once you recognise 'distance', 'direction cosines', or 'plane' in the question — meaning they're often the fastest marks in the entire chapter once the formula list (Section 9–11 of the main chapter) is memorised, similar in spirit to how Matrices & Determinants offers high marks-per-minute of revision.

The point-to-line distance formula and its 3D sibling, point-to-plane distance — they share an identical structure (|substitute|/√sum of squared coefficients) and show up across circles, tangency conditions, and multiple 3D questions. Getting this one pattern automatic pays off across several sub-topics at once.

Know that it exists, but prioritise the homogeneous, through-the-origin case (ax²+2hxy+by²=0) and its angle/perpendicularity/coincidence conditions — that version is tested far more often. The general condition is a lower-frequency addition worth a quick look, not a heavy drilling target.

Reusing a 2D formula unchanged in a 3D question — most often forgetting the z-term in the distance or section formula. It's an easy trap because the 2D version becomes so automatic that the third coordinate silently gets dropped under time pressure. Consciously write out all three coordinate differences before touching any 3D formula.