Question

The angle between the tangents drawn from the point (2, 2) to the circle \(x^2 + y^2 + 4x + 4y + c = 0\) is \(\cos^{-1} \left( \frac{7}{16} \right)\). If two such circles exist, then the sum of values of \(c\) is

• A. 16
• B. 20
• C. -20
• D. -16

Hint:

Angle between tangents can be derived using geometry and cosine relation involving distance and radius.

Answer 1

The Correct Option is D

Use formula: \[ \cos \theta = \dfrac{\sqrt{(x - a)^2 + (y - b)^2 - r^2}}{\sqrt{(x - a)^2 + (y - b)^2}} \] or apply length of tangent from external point \(= \sqrt{(x - a)^2 + (y - b)^2 - r^2}\), then relate angle with dot product or law of cosines. Sum of c values derived accordingly.