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. ` alpha^(2)+beta^(2)=(a^(2))/(2)`
B. `alpha^(2)+beta^(2)=a^(2)`
C. `alpha^(2)+beta^(2)=2a^(2)`
D. none of these

Answer

C

Explanation

The equation of the chord of contact of tangents drawn from `P (alpha , beta)` to the circle `x^(2)+y^(2)=a^(2)` is
`alpha x + beta y = a^(2) " " ...(i)`
The combined equation of OQ and OR is a homogeneous equation of second degree obtained with the help of `x^(2)+y^(2)=a^(2)`and (i) and is given by
`x^(2)+y^(2)=a^(2)((alpha x + beta y)/(a)^(2))^(2)rArr a^(2)(x^(2)+y^(2))=(alpha x + beta y)^(2)`

Lines given by this equation are mutually perpendicular.
`:.` Coeff. of `x^(2)` + Coeff. of `y^(2)=0`
`rArr alpha^(2)-a^(2)+beta^(2)-a^(2)=0 rArr alpha^(2)+beta^(2)=2a^(2)`