First slide

Definition of a circle

Question

The locus of the mid points of a chord of the circle $x^{2} + y^{2} = 4$ which subtends a right angle at the origin, is

Moderate

A

$x + y = 2$
B

$x^{2} + y^{2} = 1$
C

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

$x + y = 1$

Solution

Let $(h, k)$ be the coordinates of the mid point of a chord which subtends a right angle at the origin. Then equation of the chord is
$h x + k y - 4 = h^{2} + h^{2} - 4$ $[U \sin g T = S]$
$\Rightarrow h x + k y = h^{2} + k^{2}$
The combined equation of the pair of lines joining the origin to the points of intersection of $x^{2} + y^{2} = 4$ and $h x + k y = h^{2} + k^{2}$
$x^{2} + y^{2} - 4 {(\frac{h x + k y}{h^{2} + k^{2}})}^{2} = 0$
Lines given by the above equation are at right angle. Therefore.
$Coeff. of x^{2} + Coeff. of y^{2} = 0$
$\begin{array}{l} \Rightarrow 2 {(h^{2} + k^{2})}^{2} - (4 h^{2} + 4 k^{2}) = 0 \\ \Rightarrow h^{2} + k^{2} = 2 \end{array}$
Hence, the locus of $(h, k)$ is $x^{2} + y^{2} = 2$

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