Explanation

(d)
The tangent at (1, 7) to the parabola $x^{2} = y - 6$ is
$x = \frac{1}{2}(y + 7) - 6$
$\Rightarrow 2x = y + 7 - 12$
$\Rightarrow y = 2x + 5$
Which is also tangent to the circle
$x^{2} + y^{2} + 16x + 12y + c = 0$
$\therefore x^{2} + (2x + 5)^{2} + 16x + 12(2x + 5) + c = 0$
Or $5x^{2} + 60x + 85 + c = 0$
Must have equal roots
Let $\alpha\ and\ \beta$ are the roots of the equation
$\Rightarrow \ \alpha + \beta = - 12\ \Rightarrow \ \alpha = - 6\ \ \ \ (\because\ \alpha = \beta)$
$\therefore\ x = - 6\ and\ y = 2x + 5 = - 7$
$\Rightarrow$ point of contact is $( - 6,\ - 7)$