Question

If the angle between the circles \( x^2 + y^2 + 2x - 4y + 1 = 0 \) and \( x^2 + y^2 - 4x - 2y + c = 0 \) is \( \frac{\pi}{4} \), then \( c \) is:

• A. 3
• B. -13
• C. -3 or 13
• D. -31 or -3

Hint:

To find the angle between two circles, compare their general forms and apply the formula involving dot products or distance between centers.

Answer 1

The Correct Option is A

Angle between two circles is given by the formula: \[ \cos \theta = \frac{2\sqrt{r_1^2 r_2^2}}{d^2 - r_1^2 - r_2^2 + 2r_1 r_2 \cos \theta} \] Using the condition \( \theta = \frac{\pi}{4} \) and solving, we find that \( c = 3 \).