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 t…

Question

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:

Accepted Answer

$y=x^2+6$, $y'=2x$; at $(1,7)$, slope $2$: $y-7=2(x-1)\Rightarrow2x-y+5=0$.

Circle: centre $(-8,-6)$, $r^2=100-c$. Distance from centre to the tangent equals $r$:
$$\frac{|2(-8)-(-6)+5|}{\sqrt5}=\frac5{\sqrt5}=\sqrt5\Rightarrow r^2=5=100-c\Rightarrow c=95.$$

Solutions · 1