The locus of the centre of the circle, which cuts the circle `x^(2) + y^(2) - 20x + 4 = 0` orthogonally and touchs the line x = 2 , is

Options

A. `x^(2) = 16y`
B. `y^(2) = 4x`
C. `y^(2) = 16 x`
D. `x^(2) = 4y`

Answer

C

Explanation

Given circle is , `S = x^(2) + y^(2) - 20 x + 4 = 0` and
Line `x = 2 implies x - 2 = 0`
Let the required circle is , `S^(1) -= x^(2) + y^(2) + 2gx + 2gy + c = 0`
S = 0 cur `S^(1) = 0` orthogonally
`implies - 20 g + 0 = 4 + c [ :. 2gg, + 2ff. = c + c.]`
`implies c = - 20 g - 4`
S = 0 touch the line x = 2
`implies |g + 2| = sqrt(g^(2) + f^(2) - c) [:. c = - 20 g - 4]`
`implies g + 2 = sqrt(g^(2) + f^(2) + 20g + 4)`
`implies g^(2) + 4g = 4 = g^(2) + f^(2) + 20g + 4`
`impies f^(2) + 16 g = 0`
The locus of the centre (-g, f) is `y^(2) - 16x = 0`
`:. y^(2) = 16 x`