Question

Vertex is $(4,3)$ and the directrix is $3x+2y-7=0$, then equation of latus rectum is:

Answer 1

Vertex$=(4,3)$

Equation of directrix

$\Rightarrow 3x+2y-7=0$

$\therefore y=-\frac{3}{2}x+7$

The slope of latus rectum$=\frac{-3}{2}$

Slope of axis$=\frac{k-3}{h-4}$

$\therefore \left(\frac{k-3}{h-4}\right)\times \left(\frac{-3}{2}\right)=-1$

[$\because$ axis is perpendicular to L. R]

$\therefore 3k-9=2h-8$

$\therefore 2h-3k=1$ ………..$\left(1\right)$

SP$=2$VP

$\left|\frac{3h+2k-7}{\sqrt{9+4}}\right|=2\left|\frac{12+6-7}{\sqrt{9+4}}\right|$

$|3h+2k-7|=22$

$3h+2k=29$ …………$\left(2\right)$

$\therefore$ Solving $\left(1\right)$ & $\left(2\right)$

$h=\frac{85}{13}$

$k=\frac{61}{13}$

L.R$=y-\left(\frac{61}{13}\right)=\frac{-3}{2}(x-\frac{85}{13})$ - slope point form

$2(13y-61)=-39x+255$

$26y-122+39x-255=0$

$39x+26y-377=0$

$\Rightarrow 3x+2y-29=0$.