Question

Find the equation of the parabola whose focus is $(-2,3)$ and directrix is the line $2x+3y-4=0$. Also find the length of the Latus rectum and equation of the axis of parabola.

Answer 1

In parabola directrix is at $2a$ distance from focus

So $2a=d(P,L)$

$=\frac{|a{x}_1+b{y}_1+c|}{\sqrt{a^2+b^2}}$

$=\frac{|2\times (-2)+3\times 3-4|}{\sqrt{2^2+3^2}}=\frac{1}{\sqrt{5}}$

So $4a=\frac{2}{\sqrt{5}}$

So length of latus rectum $=4a=\frac{2}{\sqrt{5}}$units

The axis is perpendicular to directrix

For $ax+by+c=0$ perpendicular is of the form $bx-ay+k=0$

So axis is of the form $3x-2y+k=0$

As the focus lies on the axis $3(-2)-2\left(3\right)+k=0$

Hence $k=12$

So axis is $3x-2y+12=0$