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:

Let the equation of the circle through (a,b)  be x2+y2+2gx+2fy+c=0                                 …….(1)∴        a2+b2+2ga+2fb+c=0                     ………(2)If circle (1) cuts the circle x2+y2−p2=0 , orthogonally, then2g.0+2f.0=c−p2⇒c=p2 .If C(h,k)  is the centre of the circle (1), then    h=−g,k=−f⇒g=−h,f=−k .Substituting in (2), we get     a2+b2−2ah−2bk+p2=0 or      2ah+2bk−(a2+b2+p2)=0 ∴    Locus of C(h,k)  is2ax+2by−(a2+b2+p2)=0 .