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

Equation of tangent to the parabola at (1,7) is
$x-\frac{(y+7)}{2}+6=0\Rightarrow 2x-y+5=0$ 
$\Rightarrow$ centre = (-8,-6)
Equation of CQ = x + 2y + k = 0
$-8-12+k=0\Rightarrow k=20$

$$\begin{gathered} PQ=4x-2y+10=0 \\ \text{CQ=x+2y+20=0}\Rightarrow 5x+30=0\Rightarrow x=-6 \\ \Rightarrow -6+2y+20=0\Rightarrow y=-7 \end{gathered}$$

Hence, the point of contact is (-6,-7).