If the four distinct points (4,6),(-1,5),(0,0) and (k,3k) lie on a circle of radius r, the…

Question

If the four distinct points $(4,6),(-1,5),(0,0)$ and $(k,3k)$ lie on a circle of radius $r$, then $10k+r^2$ is equal to:

Accepted Answer

Circle through the origin: $x^2+y^2-Dx-Ey=0$. Through $(4,6)$: $4D+6E=52$. Through $(-1,5)$: $D-5E=-26$.

Solving: $E=6,\ D=4$. Circle: $x^2+y^2-4x-6y=0$, centre $(2,3)$, $r^2=13$.

$(k,3k)$ on the circle: $10k^2-22k=0\Rightarrow 10k=22$ (for $k
e0$).

$10k+r^2=22+13=35.$

Solutions · 1