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$A (a, 0), B (- a, 0)$ are two points. If a point P moves such that $∠ PAB - ∠ PBA = π / 2$ The locus of p is

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$x^{2} + y^{2} = a^{2}$

$x^{2} - 2 xy - y^{2} = a^{2}$

$x^{2} - y^{2} = a^{2} = 0$

$x^{2} - y^{2} = a^{2}$

answer is D.

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Detailed Solution

$tanβ = \frac{y}{a - x}, tanγ = \frac{y}{a + x} \cot (β - γ) = \cot \frac{π}{2} = 0 \Rightarrow cotβcotγ + 1 = 0 \Rightarrow (\frac{a - x}{y}) (\frac{a + x}{y}) = - 1 \Rightarrow a^{2} - x^{2} = - y^{2} \Rightarrow x^{2} - y^{2} = a^{2}$

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