Animated Solution for Mathematics - Circles: If the tangent at $(1,7)$ to the curve $x^2=y-6$ touches the circle $x^2+y^2+16x+12y+c=0$ then the value of $c$ is :

Imagine you are standing on a coordinate plane, looking at two distinct, elegant shapes: a parabola, $x^2=y-6$, and a circle, $x^2+y^2+16x+12y+c=0$.

We are looking for a specific value of $c$ that makes a tangent line to the parabola also kiss the circle perfectly. This is a story of how lines and curves interact in the beautiful world of coordinate geometry.

Our journey begins with the parabola $x^2=y-6$. We are given a point $(1,7)$.

Before we do anything else, we must verify that this point actually belongs to the parabola. Plugging in $x=1$, we get $1^2=1$, and $y-6=7-6=1$. Since $1=1$, the point is indeed on the curve.

To find the tangent, we use the $T=0$ transformation. This method allows us to write the equation of a tangent at any point $(x_1,y_1)$ by replacing $x^2$ with $x{x}_1$ and $y$ with $\frac{y+y_1}{2}$.

Applying this to our parabola, we get:

Simplifying this, we multiply by $2$ to get $2x=y+7-12$. This rearranges beautifully into the standard linear form:

This is our green line, the bridge between our two shapes.

Now, let us turn our attention to the circle $x^2+y^2+16x+12y+c=0$. To understand its relationship with our tangent line, we need to know its center and radius.

Comparing this to the general form $x^2+y^2+2gx+2fy+c=0$, we identify $2g=16$ and $2f=12$, giving us $g=8$ and $f=6$. The center is $(-g,-f)$, which is $(-8,-6)$.

The radius $r$ is given by $\sqrt{g^2+f^2-c}$, which simplifies to:

The problem states that our tangent line $2x-y+5=0$ is also a tangent to this circle. Geometrically, this means the perpendicular distance $d$ from the center $(-8,-6)$ to the line must be exactly equal to the radius $r$.

We use the distance formula:

Substituting our values, we get:

Calculating the numerator, we have $|-16+6+5|=|-5|=5$. The denominator is $\sqrt{4+1}=\sqrt{5}$. Thus:

We now equate the distance $d$ to the radius $r$:

Squaring both sides, we get $5=100-c$. Solving for $c$, we find:

The mystery is solved! By ensuring $c=95$, we have perfectly aligned the circle so that the tangent line to the parabola also kisses the circle.