First slide

Definition of a circle

Question

The locus of the middle point of chords of the circle $x^{2} + y^{2} = a^{2}$ which pass through the fixed point $(h, k),$ is

Moderate

A

$x^{2} + y^{2} - h x - k y = 0$
B

$x^{2} + y^{2} + h x + k y = 0$
C

$x^{2} + y^{2} - 2 h x - 2 k y = 0$
D

$x^{2} + y^{2} + 2 h x + 2 k y = 0$

Solution

Let $(α, β)$ be the mid point of a chord of the circle $x^{2} + y^{2} = a^{2} .$ Then its equation is
$α x + β y = α^{2} + β^{2}$ [Using $S = T$ ]
This passes through $(h, k) .$
$∴ α h + β k = α^{2} + β^{2}$
Hence, the locus of $(α, β)$ is
$x^{2} + y^{2} = h x + k y \Rightarrow x^{2} + y^{2} - h x - k y = 0 .$

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