'Circles'

Definition of a Circle

A circle is the locus of a point that moves in a plane such that its distance from a fixed point (centre) is always constant (radius).

Standard Equation of a Circle

Circle with centre (h, k) and radius r: (x - h)^2 + (y - k)^2 = r^2

Circle with Centre at Origin

x^2 + y^2 = r^2

Parametric Form

For circle x^2 + y^2 = r^2, parametric equations: x = r cos theta, y = r sin theta, where theta in [0, 2pi)

General Equation of a Circle

x^2 + y^2 + 2gx + 2fy + c = 0

Centre

C(-g, -f)

Radius

r = sqrt(g^2 + f^2 - c)

Conditions for Circle to be Real

• g^2 + f^2 - c > 0: Real circle
• g^2 + f^2 - c = 0: Point circle (radius = 0)
• g^2 + f^2 - c < 0: Imaginary circle (no real locus)

General Equation Conditions

Coefficient of x^2 = coefficient of y^2 (both 1 in standard form). No xy term (coefficient of xy = 0).

Diameter Form

Equation of a circle with A(x_1, y_1) and B(x_2, y_2) as endpoints of a diameter: (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0

Proof: For any point P(x, y) on the circle, angle APB = 90 degrees (angle in a semicircle). So slope(PA) x slope(PB) = -1.

Tangent to a Circle

Equation of Tangent

For circle x^2 + y^2 + 2gx + 2fy + c = 0 at point P(x_1, y_1): T = 0: xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0

For circle x^2 + y^2 = r^2 at (x_1, y_1): xx_1 + yy_1 = r^2

Condition of Tangency

Line y = mx + c is tangent to x^2 + y^2 = r^2 if: c^2 = r^2(1 + m^2) or c = pm r sqrt(1 + m^2)

Normal to a Circle

The normal at a point on a circle is the line perpendicular to the tangent at that point, passing through the centre of the circle.

For circle x^2 + y^2 = r^2 at (x_1, y_1): y - y_1 = y_1/x_1 (x - x_1) if x_1 != 0 (along the radius).

More generally, the normal passes through the centre (-g, -f).

Length of Tangent from an External Point

Length of tangent from P(x_1, y_1) to circle x^2 + y^2 + 2gx + 2fy + c = 0: PT = sqrt(x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c) = sqrt(S_1)

Power of a Point

For a point P(x_1, y_1) and circle S = 0:

• If S_1 > 0: P is outside the circle
• If S_1 = 0: P lies on the circle
• If S_1 < 0: P is inside the circle

Worked Examples

Example 1: Find the centre and radius of x^2 + y^2 - 4x + 6y - 12 = 0. Solution: 2g = -4 => g = -2, 2f = 6 => f = 3, c = -12. Centre = (2, -3). Radius = sqrt(4 + 9 + 12) = sqrt(25) = 5.

Example 2: Find the equation of the circle with centre (2, -3) and radius 4. Solution: (x-2)^2 + (y+3)^2 = 16 => x^2 + y^2 - 4x + 6y + 9 + 4 - 16 = 0 => x^2 + y^2 - 4x + 6y - 3 = 0.

Example 3: Find the equation of the circle whose diameter endpoints are (2, 3) and (4, 5). Solution: Using diameter form: (x-2)(x-4) + (y-3)(y-5) = 0 => x^2 - 6x + 8 + y^2 - 8y + 15 = 0 => x^2 + y^2 - 6x - 8y + 23 = 0

Example 4: Find the equation of the tangent to x^2 + y^2 = 25 at (3, 4). Solution: T = 0: 3x + 4y = 25.

Common Mistakes

1. Coefficient of x^2 and y^2 must be 1: Before finding centre and radius, divide the equation if coefficients are not 1.
2. Sign of g and f: From x^2 + y^2 + 2gx + 2fy + c = 0, centre is (-g, -f).
3. Length of tangent formula: sqrt(S_1) where S_1 is obtained by substituting the point, NOT sqrt(g^2 + f^2 - c).
4. Point is outside: Tangent exists only if the point is outside the circle (S_1 > 0).

ISC Exam Focus

• Theory (70%): Standard and general equations, centre and radius, parametric form.
• Application (30%): Finding equation given conditions, tangents, normals.
• ISC frequently asks: "Find the equation of the circle passing through three points."
• 4-6 mark questions involving tangents and circle properties.

Self-Test Questions

Q1: Find the centre and radius of x^2 + y^2 + 8x - 10y + 32 = 0. Answer: 2g = 8 => g = 4, 2f = -10 => f = -5. Centre = (-4, 5). Radius = sqrt(16 + 25 - 32) = sqrt(9) = 3.

Q2: Find the equation of the circle with centre (-1, 2) and passing through (3, 5). Answer: Radius = sqrt((3+1)^2 + (5-2)^2) = sqrt(16+9) = 5. Equation: (x+1)^2 + (y-2)^2 = 25. => x^2 + y^2 + 2x - 4y + 1 + 4 - 25 = 0 => x^2 + y^2 + 2x - 4y - 20 = 0.

Q3: Find k if the circle x^2 + y^2 + 4x + 6y + k = 0 has radius 3. Answer: g = 2, f = 3. r^2 = g^2 + f^2 - c = 4 + 9 - k = 9. So k = 4.

Q4: Find the equation of the tangent to x^2 + y^2 = 13 at (2, -3). Answer: 2x - 3y = 13.

Q5: Find the equation of the circle passing through (1, 1), (2, -1), and (3, 2). Answer: Let equation be x^2 + y^2 + 2gx + 2fy + c = 0. Substitute points and solve the three equations for g, f, c.

Q6: Find the length of the tangent from (4, 5) to the circle x^2 + y^2 - 2x + 4y - 20 = 0. Answer: S_1 = 16 + 25 - 8 + 20 - 20 = 33. Length = sqrt(33).