Find the condition that the chord of contact of tangents from the point `(alpha, beta)` to the circle `x^2 + y^2 = a^2` should subtend a right angle at the centre. Hence find the locus of `(alpha, beta)`.

Options

A. 2a
B. a/2
C. `sqrt(2)a`
D. `a^(2)`

Answer

C

Explanation

Let P(h,k) be the point. The, the chord of contact of tangents drawn from P to the circle `x^(2)+y^(2)=a^(2)`is `hx+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)((hx+ky)/(a^(2)))^(2)rArr 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`
`rArr h^(2)-a^(2)+k(2)-a^(2)=0rArr 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`.