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Question

A circle S passes through the point (0, 1) and is orthogonal to the circles (x1)2+y2=16 and x2+y2=1 then,

A
Radius of S is 8
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B
Radius of S is 7
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C
Centre of S is (-7, 1)
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D
Center of S is (-8, 1)
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Solution

The correct option is C Centre of S is (-7, 1)
(a)The general equation of a circle is
x2+y2+2gx+2fy+c=0
Where, centre and radius are given by (g,f) and g2+f2c, respectively.

(b) If the two circles x2+y2+2g1x+2f1y+c1=0 and x2+y2+2g2x+2f2y+c2=0 are orthogonal, then 2g1g2+2f1f2=c1+c2.

Let the circle be x2+y2+2gx+2fy+c=0
It passes through (0, 1).
1+2f+c=0 ....(i)
Orthogonal with
x2+y22x15=02g(1)=c15 using property (b) stated above
c=152g ....(ii)
Similarly,
Orthogonal with x2+y21=0
therefore c=1 .....(iii)
Solving (i), (ii) and (iii), we get
g=7 and f=1
Centre is (g,f)=(7,1)
Radius = g2+f2c
=49+11=7

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