Question

If the equation of the polar of the point \( (\alpha, -1) \) with respect to the circle \( x^2+y^2-4x-6y-12=0 \) is \( y = \beta \), then \( 4(\alpha+\beta) = \):

• A. \( -5 \)
• B. \( 7 \)
• C. \( -6 \)
• D. \( 0 \)

Hint:

To find the equation of the polar of a point with respect to a circle, use: \[ Dx_0 + Ey_0 + F + x x_0 + y y_0 = 0 \] This simplifies the computation efficiently.

Answer 1

The Correct Option is C

The equation of the given circle: \[ x^2 + y^2 - 4x - 6y - 12 = 0 \] Rewriting in standard form: \[ (x-2)^2 + (y-3)^2 = 25 \] The equation of the polar of a point \( (x_0, y_0) \) with respect to a circle \( x^2 + y^2 + Dx + Ey + F = 0 \) is: \[ Dx_0 + Ey_0 + F + x x_0 + y y_0 = 0 \] Substituting \( (\alpha, -1) \): \[ -4\alpha - 6(-1) - 12 + x\alpha - y(-1) = 0 \] Simplifying: \[ -4\alpha + 6 - 12 + x\alpha + y = 0 \] \[ x\alpha + y = 4\alpha - 6 \] Since \( y = \beta \), equating: \[ \beta = 4\alpha - 6 \] Thus, \[ 4(\alpha + \beta) = 4\left(\alpha + (4\alpha - 6) \right) \] \[ = 4(5\alpha - 6) \] \[ = -6 \]