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AP EAMCET · Maths · Circle

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

  1. A \(\left(-1, \frac{3}{2}\right)\)
  2. B \(\left(\frac{3}{2},-1\right)\)
  3. C \(\left(\frac{1}{2}, \frac{-3}{2}\right)\)
  4. D \(\left(1, \frac{-3}{2}\right)\)
Verified Solution

Answer & Solution

Correct Answer

(D) \(\left(1, \frac{-3}{2}\right)\)

Step-by-step Solution

Detailed explanation

Center of given circle \((C)\): \(C = \left(\frac{-(-4)}{2}, \frac{-(-6)}{2}\right) = (2, 3)\) PAB circle has PC as diameter. Center of PAB circle \(= \text{Midpoint of } PC\) \((-g, -f) = \left(\frac{-4+2}{2}, \frac{0+3}{2}\right) = \left(-1, \frac{3}{2}\right)\)…