First slide

Definition of a circle

Question

If the chord of contact of tangents drawn from the point (h, k) to the circle $x^{2} + y^{2} = a^{2}$ subtends a right angle at the centre, then

Moderate

A

$h^{2} + k^{2} = a^{2}$
B

$2 (h^{2} + k^{2}) = a^{2}$
C

$h^{2} - k^{2} = a^{2}$
D

$h^{2} + k^{2} = 2 a^{2}$

Solution

The equation of the chord of contact of tangents drawn from the point $(h, k)$ to the circle $x^{2} + y^{2} = a^{2}$ is $h x + k y = a^{2}$ The combined equation of $O Q$ and $O R$ is
$x^{2} + y^{2} = a^{2} {(\frac{h x + k y}{a^{2}})}^{2}$
Since $O Q$ is perpendicular to $O R$ . Therefore,
$Coeff. of x^{2} + Coeff. of y^{2} = 0 \Rightarrow 2 a^{2} = h^{2} + k^{2}$

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