The limiting points of the coaxial system containing the two circles `x^(2)+y^(2)+2x-2y+2=0` and `25(x^(2)+y^(2))-10x-80y+65=0` are

Options

A. (1, -1), (-5, -40)
B. (1, -1), (-1/5, -8/5)
C. (-1, 1) , (1/5, 8/5)
D. (-1, 1), (-1/5, -8/5)

Answer

C

Explanation

The equation of the radical axis of the given circles is `4x+2y-1=0`. Therefore, the equation of family of coaxial circles is
`x^(2)+y^(2)+2x-2y+2+lambda(4x+2y-1)=0`
`rArr x^(2)+y^(2)+2x(1++2lambda)+2y(lambda-1)+2-lambda=0`
The coordinates of the centre and radius of this circle are
`-(2lambda+1), -(lambda-1) and sqrt((2lambda+1)^(2)+(lambda-1)^(2)-2)`
For limiting points, we must have
Radius=0
`rArr (2lambda+1)^(2)+(lambda-1)^(2)+lambda-2=0 rArr lambda=0, -3//5`
Hence, the limiting points are (-1, 1) and (1/5, 8/5)