Question

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

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is C $6$

Given equation of parabola is

$y^2=3(2x-3)$

$\Rightarrow y^2=6(x-\frac{3}{2})$

Vertex is $(\frac{3}{2},0)$

Focus is $S(3,0)$ ($y^2=4ax$. Focus is $(a,0))$

Equation of directrix is:

$x-\frac{3}{2}=-\frac{3}{2}$, i.e., $x=0$

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

So, the coordinates of $M$ are $(0,3t)$

$\Rightarrow MS=\sqrt{9+9t^2}$

$P{S}^2=(-\frac{3}{2}+\frac{3}{2}{t}^2{)}^2+(3t{)}^2$

For parabola SM=PM

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

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

$\therefore 2=-1+t^2$

$t^2=3$

$\Rightarrow SM=6=PS$

$M=(0,3\sqrt{3})$,$P=(6,3\sqrt{3})$

PM=6

Hence, length of side of $△SMP=6$.