The locus of the centre of a circle which cuts a given circle orthogonally and also touches a given straight line is (a) circle (c) parabola (b) line (d) ellipse

Options

A. circle
B. line
C. parabola
D. ellipse

Answer

C

Explanation

Without the loss of generality, the circle can be taken as `x^(2) + y^(2) = a^(2)` and the given line as `x =b =0`.
Let (h,k) be the centre of the required circle. Then length of tangent from (h,k) to the circle and distance of (h,k) from the line should be equal.
Hence, `sqrt(h^(2) + k^(2) -a^(2)) = |h-b|` or `k^(2) -a^(2) =- 2bh +b^(2)`
Thus, the locus is `y^(2) - a^(2) =- 2bx +b^(2)`.
i.e, `y^(2) =- 2bx +b^(2) +a^(2)` which is a parabola.