First slide

Definition of a circle

Question

The locus of the middle points of chords of the
circle $x^{2} + y^{2} = 25$ which are parallel to the line $x - 2 y + 3 = 0$ is

Moderate

A

$x + 2 y = 0$
B

$2 x + y = 0$
C

$x - 2 y = 0$
D

$2 x - y = 0$

Solution

Let $(h, k)$ be the mid-point of a chord of the circle
$x^{2} + y^{2} = 25$ Then, equation of the chord is
$h x + k y - 25 = h^{2} + k^{2} - 25 \Rightarrow h x + k y = h^{2} + k^{2}$
If this is parallel to $x - 2 y + 3 = 0,$ then
$- \frac{h}{k} = \frac{1}{2} \Rightarrow 2 h + k = 0$
So, the locus of $(h, k)$ is $2 x + y = 0$

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