JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Parabola

The sides $x=0$ and $y=3$ meet at a right angle at $(0,3)$, so the triangle is right-angled and its circumcentre is the midpoint of the hypotenuse, which is the third side. A tangent to $y^2=6x$ is $y=mx+\frac{3}{2m}$, cutting $x=0$ at $\left(0,\frac{3}{2m}\right)$ and $y=3$ at $\left(\frac{6m-3}{2m^2},3\right)$. The circumcentre $(h,k)$ is the midpoint of these two points, so $2h=\frac{6m-3}{2m^2}$ and $2k-3=\frac{3}{2m}$. Eliminating $m$ gives $4k^2-18k+3h+18=0$, so the locus is $4y^2-18y+3x+18=0$.