Mathematics - Parabola Question with Solution | TestHub

The curve is $x^2=y-6$. Differentiating with respect to $x$, we get $2x=\frac{dy}{dx}$.

At $(1,7)$, the slope of the tangent is $\frac{dy}{dx}=2(1)=2$.

The equation of the tangent is $y-7=2(x-1)$, which simplifies to $2x-y+5=0$.

The circle is $x^2+y^2+16x+12y+c=0$. Its center is $(-8,-6)$ and radius is $r=\sqrt{(-8{)}^2+(-6{)}^2-c}=\sqrt{64+36-c}=\sqrt{100-c}$.

Since the tangent touches the circle, the perpendicular distance from the center of the circle to the tangent line equals the radius.

Distance $d=\frac{|2(-8)-(-6)+5|}{\sqrt{2^2+(-1{)}^2}}=\frac{|-16+6+5|}{\sqrt{4+1}}=\frac{|-5|}{\sqrt{5}}=\frac{5}{\sqrt{5}}=\sqrt{5}$.

So, $\sqrt{100-c}=\sqrt{5}$. Squaring both sides, $100-c=5$, which gives $c=95$.