Question

If $P$ is a point on the parabola $y^2=3(2x-3)$ and $M$ is foot of perpendicular drawn from $P$ on the directrix of the parabola, then the length of each side of an equilateral triangle $SMP$, where $S$ is focus of the parabola is _____________.

Answer 1

Equation of the parabola is $y^2=3(2x-3)\Rightarrow y^2=6(x-\frac{3}{2})$

Therefore equation of directrix is $x-\frac{3}{2}=-\frac{3}{2}\Rightarrow x=0$

Let the coordinate of $P$ be $(\frac{3}{2}+\frac{3}{2}{t}^2,3t)$

Coordinates of $M$ are $(0,3t)$

$\therefore MS=\sqrt{9+9t^2}$ and $MP=\frac{3}{2}+\frac{3}{2}{t}^2$

But $MS=MP$

$\Rightarrow 9+9t^2={(\frac{3}{2}+\frac{3}{2}{t}^2)}^2$

$\Rightarrow 9(1+t^2)=\frac{9}{4}(1+t^2{)}^2$

$\Rightarrow 4=1+t^2$

$\Rightarrow t^2=3$

Therefore length of the side $=\frac{3}{2}+\frac{3}{2}{t}^2=\frac{3}{2}+\frac{3}{2}\left(3\right)=\frac{3}{2}+\frac{9}{2}=\frac{12}{2}=6$

$\therefore$ length of the side$=6$.