Step 1: Understanding the Concept:
First, find the equation of the tangent line to the given parabola at the specified point.
Second, if a line touches a circle (is a tangent to it), the perpendicular distance from the center of the circle to the line is equal to the radius of the circle.
Step 2: Key Formula or Approach:
1. Equation of tangent at $(x_1, y_1)$ can be found using derivative $m = y'(x_1)$ and $y-y_1 = m(x-x_1)$.
2. For circle $x^2 + y^2 + 2gx + 2fy + c = 0$, center is $(-g, -f)$ and radius $r = \sqrt{g^2 + f^2 - c}$.
3. Perpendicular distance from point $(x_0, y_0)$ to line $Ax+By+C=0$ is $d = \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$. Equate $d=r$.
Step 3: Detailed Explanation:
Let's find the tangent to the curve $x^2 = y - 6$ at point $(1, 7)$.
Differentiate with respect to $x$:
\[ 2x = \frac{dy}{dx} \]
Evaluate the slope at $x = 1$:
\[ m = \left. \frac{dy}{dx} \right|_{x=1} = 2(1) = 2 \]
The equation of the tangent line using point-slope form:
\[ y - 7 = 2(x - 1) \]
\[ y - 7 = 2x - 2 \]
\[ 2x - y + 5 = 0 \quad \dots \text{ (Tangent Line)} \]
Now, consider the circle equation: $x^2 + y^2 + 16x + 12y + C = 0$.
Comparing with standard form, $2g = 16 \implies g = 8$ and $2f = 12 \implies f = 6$.
Center of the circle is $(-g, -f) = (-8, -6)$.
The radius of the circle is $r = \sqrt{g^2 + f^2 - C} = \sqrt{8^2 + 6^2 - C} = \sqrt{64 + 36 - C} = \sqrt{100 - C}$.
The line $2x - y + 5 = 0$ is tangent to the circle, so the perpendicular distance from the center $(-8, -6)$ to the line is equal to the radius $r$.
\[ d = \frac{|2(-8) - 1(-6) + 5|}{\sqrt{2^2 + (-1)^2}} \]
\[ d = \frac{|-16 + 6 + 5|}{\sqrt{4 + 1}} = \frac{|-5|}{\sqrt{5}} = \frac{5}{\sqrt{5}} = \sqrt{5} \]
Equate distance to radius:
\[ \sqrt{100 - C} = \sqrt{5} \]
Square both sides:
\[ 100 - C = 5 \]
\[ C = 100 - 5 = 95 \]
Step 4: Final Answer:
The value of C is 95.