The tangent at $(1,7)$ to the curve $x^2=y-6$ touches the circle $x^2+y^2+16x+12y+c=0$ at...

Because the line "touches" the circle, there is only one point of intersection with the circle. Therefore, the equation, $$ x^2+(2x+5)^2+16x+12(2x+5)+c=0 \implies 5 x^2 + 60 x + 85 + c= 0 $$ has one solution.

So the determinant $\Delta = 3600 - 4\cdot5\cdot(85+c)=0$. Ergo, $c=95$.

And solving the equation $5x^2 + 60x +180=0$ gives the solution $(-6,-7)$.

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Conic Sections

Analytic Geometry

Tangent Line

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