If a circle C passing through the point (4,0) touches the circle x^2+y^2+4x-6y=12 external…

Question

If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2+y^2+4x-6y=12$ externally at the point $(1,-1)$, then the radius of $C$ is:

Accepted Answer

Given circle: centre $(-2,3)$, $r=5$. Unit vector from $(-2,3)$ toward $(1,-1)$: $\left(\dfrac35,-\dfrac45\right)$.

Centre of $C$: $\left(1+\dfrac{3r_C}5,-1-\dfrac{4r_C}5\right)$, equidistant ($r_C$) from $(1,-1)$ and $(4,0)$:
$$\left(-3+\frac{3r_C}5\right)^2+\left(-1-\frac{4r_C}5\right)^2=r_C^2\ \Longrightarrow\ 10-2r_C=0\ \Longrightarrow\ r_C=5.$$

Solutions · 1