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

Question

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

Accepted Answer

$$\begin{aligned}
2 r & =\sqrt{52} \
4 r^2 & =52
\end{aligned}$$

$$\begin{aligned}
&\begin{aligned}
& 2 r^2=26 \
& r^2=13
\end{aligned}\
\&\text { Eq }{ }^{\mathrm{n}} \text { of circle is }\
\&\begin{aligned}
& x(x-4)+y(y-6)=0 \
& k(k-4)+3 k(3 k-6)=0 \
& k^2-4 k+9 k^2-18 k=0 \
& 10 k^2=22 k \
& 10 k=22 \
& \therefore \quad 10 k+r^2=35
\end{aligned}
\end{aligned}$$

Solutions · 1