The locus of the centre of the circle passing through the intersection of the circles `x^2+y^2= 1` and `x^2 + y^2-2x+y=0` is

Options

A. `x+2y=0`
B. `2x-y=1`
C. a circle
D. a pair of lines

Answer

A

Explanation

The equation of a family of circles passing through the intersection of given circles is
`x^(2)+y^(2)-2x+y+lambda(2x-y-1)=0`
or, `x^(2)+y^(2)-2(1-lambda)+(1-lambda)y-lambda=0`
Let C(h, k) be its centre. Then,
`h=1-lambda and k=-((1-lambda)/(2))rArr h=-2k`
Hence, the locus of (h, k) is `x=-2y ` or, x+2y=0