Question

From a point \( P(-4, 0) \), two tangents are drawn to the circle \( x^2+y^2-4x-6y-12=0 \) touching the circle at \( A \) and \( B \). If the equation of the circle passing through \( P, A, B \) is \( x^2+y^2+2gx+2fy+c=0 \), then \( (g,f) = \):

• A. \( (-1, \frac{3}{2}) \)
• B. \( (\frac{3}{2}, -1) \)
• C. \( (1, -\frac{3}{2}) \)
• D. \( (1, \frac{3}{2}) \)

Hint:

For problems involving circles passing through tangent points, first rewrite the given circle equation in standard form and solve accordingly.

Answer 1

The Correct Option is D

The given circle equation: \[ x^2 + y^2 - 4x - 6y - 12 = 0 \] Rewriting in standard form: \[ (x - 2)^2 + (y - 3)^2 = 25 \] Using the condition that the new circle passes through \( P \), compute \( g \) and \( f \): \[ g = 1, f = \frac{3}{2} \] Thus, the correct answer is \( (1, \frac{3}{2}) \).