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:
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:
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To find the angle between two circles, compare their general forms and apply the formula involving dot products or distance between centers.
Updated On: May 15, 2025
- 3
- -13
- -3 or 13
- -31 or -3
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The Correct Option is A
Solution and Explanation
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 \( c = 3 \).
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