If the tangent from a point p to the circle `x^2+y^2=1` is perpendicular to the tangent from p to the circle `x^2 +y^2 = 3` , then the locus of p is

Options

A. a circle of radius 2
B. a circle of radius 4
C. a circle of radius 3
D. none of these

Answer

A

Explanation

The equation of tangent to `x^(2)+y^(2)=1` is
` x cos alpha + y sin alpha = 1 " " ...(i)`
The equation of the tangent to `x^(2)+y^(2)=3`, perpendicular to (i), is
`x sin alpha - y sin alpha = sqrt(3) " " (ii)`
Let the coordinates of P be (h, k). Then,
`h cos alpha + k sin alpha = 1 ` and `h sin alpha - k cos alpha = sqrt(3)`
Eliminating (h, k) from these two equations, we get
`h^(2) + k^(2)=4`
Hence, the locus of (h, k) is `x^(2)+y^(2)-4`, which is a circle of radius 2.