The equation 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

Since x=0,y=3 are two sides  of  the triangle  given triangle  is a right angled

L is tangent to $y^2=6x$
$4a=6\Rightarrow a=\frac{3}{2}$
Tangent$\mathrm{y}=\mathrm{mx}+\frac{\mathrm{a}}{\mathrm{m}}\Rightarrow \mathrm{y}=\mathrm{mx}+\frac{3}{2\mathrm{m}}$
$\mathrm{B}=\left(0,\frac{3}{2\mathrm{m}}\right),\mathrm{C}=\left(\frac{3}{\mathrm{m}}-\frac{3}{2{\mathrm{m}}^2},3\right)$

Let $\left(x_1, y_1\right)$  be the circum centre of $\mathrm{△}\mathrm{ABC}$= mid point of hypotenuse BC

${\mathrm{x}}_1=\frac{0+\frac{3}{\mathrm{m}}-\frac{3}{2{\mathrm{m}}^2}}{2},{\mathrm{y}}_1=\frac{\frac{3}{2\mathrm{m}}+3}{2}$

${\mathrm{x}}_1=\frac{3}{2\mathrm{m}}-\frac{3}{4{\mathrm{m}}^2},2y_1-3=\frac{3}{2\mathrm{m}}$

$x_1+3-2y_1=-\frac{3}{4{\mathrm{m}}^2}$

$3x_1+9-6y_1=-{\left(2y_1-3\right)}^2$

$\begin{array}{l}\Rightarrow 4{\mathrm{y}}^2-18\mathrm{y}+3\mathrm{x}+18=0\end{array}$