Question

If a circle passes through (1, 2) and cuts the circle x² + y² = 4 orthogonally, then the equation of the locus of its centre is

(1) 2x + 4y – 9 = 0 

(2) 2x + 4y + 9 = 0 

(3) 2x – 4y + 9 = 0

(4) None of these

Answer 1

(1) 2x + 4y – 9 = 0 

Explanation

Let the circle be x² + y² + 2gx + 2fy + c = 0. This passes through (1, 2) and cuts the circle x² + y² = 4 orthogonally, therefore,

5 + 2g + 4f = c ..... (1)

and 0 = c – 4 ..... (2)

Eliminating c from (1) and (2), we get 2g + 4f + 9 = 0. Hence, the locus of the centre (–g, –f) is

– 2x – 4y + 9 = 0 or 2x + 4y – 9 = 0