The equations of two sides of a variable triangle are x=0 and y=3, and its third side is a…

Question

The equations of two sides of a variable triangle are $x=0$ and $y=3$, and its third side is a tangent to the parabola $y^2=6x$. The locus of its circumcentre is:

Accepted Answer

Third side of triangle

$t y=x+\frac{3}{2} t^{2}$

$$
\begin{aligned}
& \therefore H \equiv(0,3) \quad G \equiv\left(t-\frac{3 t}{2}\right)
\end{aligned}
$$

Let O (h, k)

$$
\begin{aligned}
& \Rightarrow \frac{2 h}{3}=t-\frac{t^{2}}{2} & \frac{2 k+3}{3}=2+\frac{t}{2} \\
& \Rightarrow 4 h=6 t-3 t^{2} & 4 k=6+3 t \\
& \Rightarrow 4 h=2(4 k-6)-3\left(\frac{(4 k-6)^{2}}{9}\right) \\
& \Rightarrow 4 h=6 k-9-\left(4 k^{2}+9-12 k\right) \\
& \Rightarrow 4 k^{2}-18 k+3 h+18=0 \\
& \Rightarrow 4 y^{2}-18 y+3 x+18=0
\end{aligned}
$$

Solutions · 1