Question

Find the coordinates of the point of intersection of the axis and the directrix of the parabola whose focus is $(3,3)$ and directrix is $3x-4y=2$. Find also the length of the latus-rectum.

Answer 1

Parabola with focus : $(3,3)$

Directrix ; $3x-4y=2................\left(1\right)$

$=>y=\frac{3}{4}x-\frac{1}{2}................\left(2\right)$

Slope of Direction $=>m_1=\frac{3}{4}$

Slope of axis must be $m_2=-\frac{1}{m_1}=>m_2=\frac{-4}{3}\left(asaxisis\perp todirectrix\right)$

Equation of axis, $y=-\frac{4}{3}x+c...............\left(3\right)$

As axis passes through focus, $(3,3)$ must satisfy equation$\left(3\right)$,

$=>3=-\frac{4}{3}\left(3\right)+c$

$=>c=7$

Then equation of axis is, $3y+4x-21=0.................\left(4\right)$

Point of intersection of axis and directrix, equation $\left(2\right)$ and $\left(4\right)$,

$M=>(\frac{18}{5},\frac{11}{5}).......................\left(5\right)$

$LR=2\times ($distance between point of intersection of axis and directrix and focus$)$

$=2\times \sqrt{(\frac{18}{5}-3{)}^2+(\frac{11}{5}-3{)}^2}$

$=2\times \sqrt{\frac{1}{25}(9+16)}$

$=2\times \sqrt{\frac{25}{25}}$

$=2unitlength.$