Introduction to Conic Sections
Conic sections are curves obtained by intersecting a double-napped right circular cone with a plane. The three main types are parabola, ellipse, and hyperbola, depending on the angle of intersection.
Types of Conics
- Parabola: Plane is parallel to a generator of the cone (eccentricity = 1).
- Ellipse: Plane cuts through the cone making an angle between 0 and the semi-vertical angle (eccentricity < 1).
- Hyperbola: Plane cuts both nappes of the cone (eccentricity > 1).
Degenerate Conics
A point, a line, or two intersecting lines (when plane passes through the vertex).
Parabola
A parabola is the locus of a point whose distance from a fixed point (focus) equals its distance from a fixed line (directrix).
Standard Equations
| Equation | Focus | Directrix | Axis | Vertex | Latus Rectum |
|---|---|---|---|---|---|
y^2 = 4ax | (a, 0) | x = -a | x-axis | (0, 0) | 4a |
y^2 = -4ax | (-a, 0) | x = a | x-axis | (0, 0) | 4a |
x^2 = 4ay | (0, a) | y = -a | y-axis | (0, 0) | 4a |
x^2 = -4ay | (0, -a) | y = a | y-axis | (0, 0) | 4a |
Key Terms
- Latus Rectum: Chord passing through focus perpendicular to axis. Length =
|4a|. - Focal Distance: Distance of any point
P(x, y)from the focus(a, 0)=|x + a|.
Parametric Form
For y^2 = 4ax: x = at^2, y = 2at, where t is parameter.
Ellipse
An ellipse is the locus of a point whose sum of distances from two fixed points (foci) is constant.
Standard Equation
x^2/a^2 + y^2/b^2 = 1 where a > b (major axis along x-axis).
Key Terms
- Major axis: length =
2a, along x-axis. - Minor axis: length =
2b, along y-axis. - Foci:
(pm c, 0)wherec^2 = a^2 - b^2. - Eccentricity:
e = c/a, where0 < e < 1. - Directrices:
x = pm a/e. - Latus Rectum: Length =
2b^2/a.
Ellipse with Vertical Major Axis
x^2/b^2 + y^2/a^2 = 1 where a > b (major axis along y-axis).
Foci: (0, pm c), eccentricity: e = c/a.
Parametric Form
x = a cos theta, y = b sin theta
Focal Property
For any point P on ellipse, PF_1 + PF_2 = 2a.
Hyperbola
A hyperbola is the locus of a point whose absolute difference of distances from two fixed points (foci) is constant.
Standard Equation
x^2/a^2 - y^2/b^2 = 1
Key Terms
- Transverse axis: length =
2a, along x-axis. - Conjugate axis: length =
2b, along y-axis. - Foci:
(pm c, 0)wherec^2 = a^2 + b^2. - Eccentricity:
e = c/a, wheree > 1. - Directrices:
x = pm a/e. - Latus Rectum: Length =
2b^2/a.
Hyperbola with Vertical Transverse Axis
y^2/a^2 - x^2/b^2 = 1
Parametric Form
x = a sec theta, y = b tan theta
Comparison Table
| Property | Parabola | Ellipse | Hyperbola |
|---|---|---|---|
| Eccentricity | e = 1 | 0 < e < 1 | e > 1 |
| Equation | y^2 = 4ax | x^2/a^2 + y^2/b^2 = 1 | x^2/a^2 - y^2/b^2 = 1 |
| Focus | One | Two | Two |
| Directrix | One | Two | Two |
| Sum/Diff of distances | Equal | Sum = const | Diff = const |
Worked Examples
Example 1: Find the focus, directrix, and latus rectum of y^2 = 16x.
Solution: Comparing with y^2 = 4ax: 4a = 16 => a = 4.
Focus = (4, 0). Directrix: x = -4. Latus rectum = 4a = 16.
Example 2: Find the equation of the ellipse with foci (pm 4, 0) and a = 5.
Solution: c = 4, a = 5. b^2 = a^2 - c^2 = 25 - 16 = 9. b = 3.
Equation: x^2/25 + y^2/9 = 1.
Example 3: Find the eccentricity of the hyperbola x^2/9 - y^2/16 = 1.
Solution: a^2 = 9 => a = 3, b^2 = 16 => b = 4. c^2 = 9 + 16 = 25 => c = 5.
e = c/a = 5/3.
Common Mistakes
- Sign of
ainy^2 = 4ax:ais positive for right-opening, negative for left-opening parabola. - Ellipse:
ais always the semi-major axis: Forx^2/a^2 + y^2/b^2 = 1,a > bmeans major axis is along x-axis. - Hyperbola:
c^2 = a^2 + b^2: Unlike ellipse wherec^2 = a^2 - b^2. - Directrix position: For
y^2 = 4ax, directrix isx + a = 0, notx - a = 0.
ISC Exam Focus
- Theory (70%): Standard equations, eccentricity, foci, directrix, latus rectum definitions.
- Application (30%): Finding equation given conditions, parametric coordinates.
- Typical ISC: "Find the equation of the parabola with focus at (a, 0) and directrix x + a = 0."
- Comparing ellipse and hyperbola equations is a common source of confusion.
Self-Test Questions
Q1: Find the focus and directrix of x^2 = -8y.
Answer: 4a = 8 => a = 2. Since negative, opens downward. Focus = (0, -2). Directrix: y = 2.
Q2: Find the equation of the parabola with focus (5, 0) and directrix x = -5.
Answer: a = 5. Equation: y^2 = 20x.
Q3: Find the eccentricity of the ellipse x^2/25 + y^2/9 = 1.
Answer: a = 5, b = 3. c^2 = 25 - 9 = 16, c = 4. e = 4/5 = 0.8.
Q4: Find the latus rectum of the hyperbola x^2/16 - y^2/9 = 1.
Answer: b^2 = 9, a = 4. Length = 2b^2/a = 18/4 = 9/2.
Q5: Find the equation of the ellipse with a = 10 and foci at (pm 6, 0).
Answer: c = 6. b^2 = a^2 - c^2 = 100 - 36 = 64. x^2/100 + y^2/64 = 1.
Q6: Find the equation of the hyperbola with foci (pm 5, 0) and transverse axis length 8.
Answer: 2a = 8 => a = 4. c = 5. b^2 = c^2 - a^2 = 25 - 16 = 9. x^2/16 - y^2/9 = 1.
