y 2 = 16x

Let the general equation of circle be

x 2 +y 2 + 2gx + 2fy + c = 0.......(i)

It cuts the circle x 2 + y 2 - 20x + 4 = 0 orthogonally

2 (-10g + 0 x f) = c + 4 20g = c + 4 ....... (ii)

circle (i) touches x = 2

therefore, perpendicular distance from centre to the tangent to the circle=radius

(g + 2) 2 = g 2 +f 2 -c

g 2 + 4 + 4g = g 2 + f 2 - c 4g + 4 = f 2 - c ... (iii)

on eliminating c from (ii) and (iii) we get

-16g + 4 = f 2 + 4 f 2 + 16g = 0

Hence, locus of (-g, - f) is,

y 2 -16x = 0 (replacing -f & - g by x &y )