Question

A circle S passes through the point (0, 1) and is orthogonal to the circles `(x -1)^2 + y^2 = 16` and `x^2 + y^2 = 1`. Then (A) radius of S is 8 (B) radius of S is 7 (C) center of S is (-7,1) (D) center of S is (-8,1)
A. radius of S is 8
B. radius of S is 7
C. centre of S is (-7, 1)
D. centre of S is (-8, 1)

Answer 1

Correct Answer - B::C
Let the circle S be `x^(2)+y^(2)+2gx+2fy+c=0`. It is orthogonal to the circles `x^(2) +y^(2)-2x-15=0` and `x^(2)+y^(2)-1=0`. Therefore,
`2(gxx(-1)+fxx0)=c-15 and 2(gxx0+fxx0)=c-1`
`rArr -2g=c-15 and c=1`
`rArr c=1, g=7`
The circle S passes through (0, 1).
`:.` The circle S passes through (0, 1)
`:. 1+2f+c=0 rArr f=-1`
Thus, the centre of the circle S is (-7, 1) and radius is `sqrt(49+1-1)=7`. So, options (b) and (c) are correct.