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:
Step-by-Step Solution
Key Concept: Parametric tangent to $y^2 = 6x$ is $y = mx + \frac{3}{2m}$. Find the three vertices of the triangle, then compute circumcentre $(h,k)$ in terms of $m$, and eliminate $m$.
Tangent to $y^2=6x$: $y = mx + \frac{3}{2m}$. Vertices: on $x=0$: $(0, \frac{3}{2m})$; on $y=3$: $(\frac{6m-3}{2m^2}, 3)$; on $x=0$ and $y=3$: $(0,3)$. Circumcentre $(h,k)$: eliminating $m$ gives $4k^2-18k+3h+18=0$, i.e., $4y^2-18y+3x+18=0$.
Correct Answer: 3