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Question

If a circle passes through the point (a,b) and cuts the circle x2+y2=p2 orthogonally , then the equation of the locus of its centre is
  1. x2+y23ax4by+(a2+b2p2)=0
  2. 2ax+2by(a2b2+p2)=0
  3. x2+y22ax3by+(a2b2p2)=0
  4. 2ax+3by(a2+b2+p2)=0

A
2ax+3by(a2+b2+p2)=0
B
x2+y23ax4by+(a2+b2p2)=0
C
x2+y22ax3by+(a2b2p2)=0
D
2ax+2by(a2b2+p2)=0
Solution
Verified by Toppr

Let the equation of the circle which passes through the point (a,b) is given by,
x2+y2+2gx+2fy+c=0..........(1)
a2+b2+2ga+2fb+c=0........(2)
Now circle (1) cuts the circle x2+y2=4 orthogonally
Therefore,
2g×0+2f×0=c4
c=4.........(3)
Substituting c=4 in (2)
a2+b2+2ga+2fb+4=0........(2)
Now let the centre of the circle by (h,k)
therefore h=g and k=f
thus,
a2+b22ha2kb+4=0
replace h by x and k by y,
thus the locus of the centre is
a2+b22xa2yb+4=0
2(ax+by)(a2+b2+4)=0

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