Question

The locus of the point, the chord of contact of tangents from which to the circle $x^2+y^2=a^2$ subtends a right angle at the centre is a circle of radius

• A. $2a$
• B. $\frac{a}{2}$
• C. $\sqrt{2}a$
• D. $a^2$

Answer 1

The correct option is C $\sqrt{2}a$

Let (h, k) be the point. Then, the chord of contact of tangents drawn from P to the circle $x^2+y^2=a^{2}ishx+ky=a^2$. The combined equation of the lines joining the (centre) origin to the points of intersection of the circle $x^2+y^2=a^2$ and the chord of contact of tangents drawn from P(h, k) is a homogeneous equation of second degree given by
$x^2+y^2=a^2(\frac{hx+ky}{a^2}{)}^2\Rightarrow a^2(x^2+y^2)=(hx+ky{)}^2$
The lines given by the above equation will be perpendicular if
$Coeff.Of{x}^2+Coeff.of{y}^{2}0$
$\Rightarrow h^2-a^2+k^2-a^2=0\Rightarrow h^2-k^2=2a^2$
Hence, the locus of (h, k) is $x^2+y^2=2a^2$
Clearly, it is a circle of radius $\sqrt{2}a$