Question:
Let \( A(5,4) \) and \( B(5,-4) \) be two points. If \( P \) is a point in the coordinate plane such that \( |APB| = \frac{\pi}{4} \), then the point \( P \) lies on the curve:
Let \( A(5,4) \) and \( B(5,-4) \) be two points. If \( P \) is a point in the coordinate plane such that \( |APB| = \frac{\pi}{4} \), then the point \( P \) lies on the curve:
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For loci problems related to fixed angles, consider applying trigonometric methods, perpendicular bisectors, and cyclic quadrilateral properties for simplification.
Updated On: Jun 5, 2025
- \( x^2 + y^2 + 10x - 17 = 0 \)
- \( x^2 + y^2 - 2x - 31 = 0 \)
- \( x^2 + y^2 - 10x + 17 = 0 \)
- \( x^2 + y^2 + 2x - 31 = 0 \)
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The Correct Option is C
Solution and Explanation
The given condition suggests that point \( P \) lies on a locus such that the angle subtended at \( P \) by segment \( AB \) is constant.
Using locus methods:
1. Compute the perpendicular bisector of \( AB \).
2. Express the equation for the locus of points where the subtended angle is fixed.
After simplifying, the correct curve equation is:
\[
x^2 + y^2 - 10x + 17 = 0
\]
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