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)

Options

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

B, C

Explanation

PLAN
(i) The general equation of a circle is
`x^(2)+y^(2)+2g_(2)x+2f_(2)y+c_(2)=0 " are orthogonal, then "2g_(1)g_(2)+2f_(1)f_(2)=c_(1)+c_(2)`.
Let circle be `x^(2)+y^(2)+2gx+2fy+c=0`
It Passes through (0, 1).
`1+2f+c=0" "...(i)`
Orthogonal with `x^(2)+y^(2)-2x-15=0`
`2g(-1)=c-15`
`rArr c= 15 -2 g " "...(ii)`
Orthogonal with `x^(2)+y^(2)-1=0`
`c= 1" "...(iii)`
`rArr g= 7 and f = -1`
Centre is `(-g, -f) -=(-7,1)`
`therefore " Radius " =sqrt(g^(2)+f^(2)-c)`
`=sqrt(49+1-1)=7`