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Q.
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 $y^2=6 x$. The locus of its circumcentre is:
$ y^2=6 x \& y^2=4 a x$
$ \Rightarrow 4 a=6 \Rightarrow a=\frac{3}{2}$
$y = mx +\frac{3}{2 m } ;( m \neq 0)$
$h =\frac{6 m -3}{4 m ^2}, k =\frac{6 m +3}{4 m }$, Now eliminating $m$ and we get
$ \Rightarrow 3 h =2\left(-2 k ^2+9 k -9\right) $
$ \Rightarrow 4 y ^2-18 y +3 x +18=0$
