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  4. Tangent to the curve y=x2+6 at a point P (1,7) touches the circle x 2+ y 2+16 x +12 y + c =0 at a point Q. Then the coordinates of Q are

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Q. Tangent to the curve $y=x^{2}+6$ at a point $P (1,7)$ touches the circle $x ^{2}+ y ^{2}+16 x +12 y + c =0$ at a point $Q$. Then the coordinates of $Q$ are

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Solution:

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Equation of tangent to the parabola at $(1,7)$ is
$x-\frac{(y+7)}{2}+6=0 \Rightarrow 2 x-y+5=0$
$\Rightarrow$ Centre$ =(-8,-6) $
Equation of $C Q-x+2 y+k=0 $
$=8-12+k=0 \Rightarrow k=20$
$P Q=4 x-2 y+10=0$
$C Q=x+2 y+20=0 $
$=5 x+30=0 \Rightarrow x=-6$
$\Rightarrow -6+2 y+20-0 \Rightarrow y=-7$
Hence the point of coatact is $(-6,-7)$.