The locus of the middle points of those chord | vedclass

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(C) Let the mid-point of the chord be $C(h, k)$.Since the chord subtends a right angle at the origin $O(0, 0)$,the triangle formed by the origin and t

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$x^2 + y^2 = 2$

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(C) Let the mid-point of the chord be $C(h, k)$.Since the chord subtends a right angle at the origin $O(0, 0)$,the triangle formed by the origin and the endpoints of the chord is an isosceles right-angled triangle.The distance from the origin to the midpoint $C(h, k)$ is $d = \sqrt{h^2 + k^2}$.In the right-angled triangle formed by the origin,the midpoint,and one endpoint of the chord,the angle at the origin is $45^\circ$.Using trigonometry,$\cos(45^\circ) = \frac{d}{r}$,where $r$ is the radius of the circle.Given $r = 2$,we have $\frac{1}{\sqrt{2}} = \frac{\sqrt{h^2 + k^2}}{2}$.Squaring both sides,$\frac{1}{2} = \frac{h^2 + k^2}{4}$,which simplifies to $h^2 + k^2 = 2$.Replacing $(h, k)$ with $(x, y)$,the locus is $x^2 + y^2 = 2$.

The locus of the middle points of those chords of the circle $x^2 + y^2 = 4$ which subtend a right angle at the origin is

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