A circle passes through the origin anf has its centre on y = x. If it cuts x2 + y2 - 4x - 6y + 10 = 0 orthogonally, the equation of the circle is

Dear Student,

Let the equation of circle be x2+y2+2gx+2hy+f=0.Therefore centre of circle is -g,-h.Since centre lies on line y=x, -h=-gh=gThus equation of circle becomes x2+y2+2gx+2gy+f=0.Since the circle passes through origin,0+0+2g0+2g0+f=0f=0Hence, equation of circle becomes x2+y2+2gx+2gy=0. Since it cuts x2+y2-4x-6y+10=0 orthogonally,2g(-2)+2g-3=10-4g-6g=10-10g=10g=10-10=-1Thus required equation of circle is x2+y2-2x-2y=0. If circle x2+y2+2gx+2hy+f=0 intersects circle x2+y2+2g1x+2h1y+f1=0, then 2gg1+2hh1=f+f1.
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