Question

The equation of the parabola with focus $S\left(\frac{3}{2},0\right)$ and directrix $2x+3=0\text{is}$

• A. $y^2=4x$
• B. $y^2=5x$
• C. $y^2=6x$
• D. none of these

Answer 1

The correct option is A $y^2=4x$

Solution:

Given focus $S=(\frac{3}{2},0)$

and directrix $\to 2x+3=0$

Let $p(x,y)$ be any point on the parabola from a perpendicular $PM$ from $P$ on thr directrix

Then $SP=PM\Rightarrow S{P}^2=P{M}^2$

$\Rightarrow (x-\frac{3}{2})+y^2={\left|\frac{2x+3}{\sqrt{2^2+3^2}}\right|}^2$

$\Rightarrow x^2+\frac{9}{4}-3x+y^2=\frac{4x^2+9+12x}{13}$

$\Rightarrow 13[4x^2-12x+4y^2+9]=16x^2+36+48x$

$\Rightarrow 52x^2-156x+52y^2+117=16x^2+36+48x$

$\Rightarrow 36x^2-204x+52y^2+81=0$

The required equation of parabola $=36x^2+52y^2-204x+81=0$