A variable chord of circle $x^{2} + y^{2} = 4$ is drawn from the point $P (3, 5)$ meeting the circle at the point A and B. A point Q is taken on this chord such that $2 P Q = P A + P B$
Locus of 'Q' is

x^{2} + y^{2} + 3 x + 4 y = 0

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x^{2} + y^{2} = 36

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x^{2} + y^{2} = 16

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x^{2} + y^{2} - 3 x - 5 y = 0

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Solution

The correct option is D $x^{2} + y^{2} - 3 x - 5 y = 0$
$2 P Q = P A + P B$
$\Rightarrow P Q - P A = P B - P Q$
$\Rightarrow A Q = Q B$
$\Rightarrow$ Q is midpoint of AB.
Let Q has coordinates $(h, k)$
Then equation of chord AB is given by $T = S_{1}$
or $h x + k y - 4 = h^{2} + k^{2} - 4$
This variable chord passes through the point $P (3, 5)$
$\Rightarrow 3 h + 5 k = h^{2} + k^{2}$
$\Rightarrow x^{2} + y^{2} - 3 x - 5 y = 0$
Which is required locus.

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