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Q.
Let $x ^{2}+ y ^{2}+ Ax + By + C =0$ be a circle passing through $(0,6)$ and touching the parabola $y = x ^{2}$ at $(2,4)$. Then $A + C$ is equal to ___
$x^{2}+y^{2}+A x+B y+C=0$ is passing through $(0,6)$
$\Rightarrow 6 B + C =-36$
The tangent of the parabola $y = x ^{2}$ at $(2,4)$ is
$4 x - y -4=0 \,\,\,----(1)$
The tangent of circle $x ^{2}+ y ^{2}+ Ax + By + C =0$ at $(2,4)$ is
$(4+ A ) x +(8+ B ) y +2 A +4 B +2 C =0\,\,\,\, -----(2)$
From Equation (1) and (2)
$\frac{4+ A }{4}=\frac{8+ B }{-1}=\frac{2 A +4 B +2 C }{-4}$
$A +4 B =-36 \,\,\, ----(3)$
$3 A +4 B +2 C =-4 \,\,\, -----(4)$
From equation (3) and (4)
$A + C =16$