AP EAMCET Engineering 21 Sep 2020 Shift 1 Paper

Let an equation of circle
x2+y2+2gx+2fy+c=0 and point A(x1,y1)
Then equation of chord of contact is
Let another point B(x2,y2) through which chord (i) passes, so
Now equation of circle having AB as diameter is
By the circles
x2+y2+2gx+2fy+c=0 and circle (iii)
the 2g1g2+2f1f2
=−2g(‌

x1+x2

2

)−2f(‌

y1+y2

2

)
=−gx1−gx2−fy1−fy2
=x1x2+y1y2+c  { from Eq. (ii) }
=c1+c2
∵2g1g2+2f1f2=c1+c2
∴ Circles x2+y2+2gx+2fy+c=0 and Eq. (iii) cuts orthogonally.