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  4. The locus of the mid-points of chords of the circle x2 + y2 = 4 which subtends a right angle at the origin is

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Q. The locus of the mid-points of chords of the circle $ x^2 + y^2 = 4$ which subtends a right angle at the origin is

IIT JEEIIT JEE 1984Conic Sections

Solution:

We have to find locus of mid-point of chord and we know perpendicular from centre bisects the chord.
$\therefore \angle O A C=45^{\circ}$
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In $\triangle O A C, \frac{O C}{O A}=\sin 45^{\circ} \Rightarrow O C=\frac{2}{\sqrt{2}}=\sqrt{2}$
Also, $\sqrt{h^{2}+k^{2}}=O C^{2}$
Hence, $x^{2}+y^{2}=2$ is required equation of locus of mid-point of chord subtending right angle at the centre.