The locus of the centre of a circle which cuts the circle x 2 - 20 x + y 2 + 4 = 0 orthogonally and also touches the line x = 2 is

Step-by-step solution

Let the general equation of circle be x 2 + y 2 + 2 g x + 2 f y + c = 0 ....... (i) It cuts the circle x 2 + y 2 - 20 x + 4 = 0 orthogonally ∴ 2 - 10 g + 0 × f = c + 4 ⇒ - 20 g = c + 4 .......(ii) ∵ circle (i) touches x = 2 therefore, perpendicular distance from centre to the tangent to the circle=radius ⇒ - g + 0 - 2 1 2 + 0 2 = g 2 + f 2 - c ⇒ g + 2 2 = g 2 + f 2 - c ⇒ g 2 + 4 + 4 g = g 2 + f 2 - c ⇒ 4 g + 4 = f 2 - c ... (iii) on eliminating c from (ii) and (iii) we get - 16 g + 4 = f 2 + 4 ⇒ f 2 + 16 g = 0 Henc