Question

Equation of the ellipse with focus $(3,-2)$,
eccentricity $\frac{3}{4}$ and directrix $2x-y+3=0$ is

• A. $44x^2+36xy+71y^2-374x-528y+756=0$
• B. $44x^2+36xy+71y^2-588x+374y+959=0$
• C. $44x^2+36xy+71y^2-125x-274y+659=0$
• D. $44x^2+36xy+71y^2-135x-47y+859=0$

Answer 1

The correct option is B $44x^2+36xy+71y^2-588x+374y+959=0$

Given that: Focus is $(3,-2)$, eccentricity is $\frac{3}{4}$ and directrix is $2x-y+3=0$ Let a point on ellipse is $P(x,y)$
equation of ellipse is the locus of point $P(x,y)$.
Now, equation is :
Distance of $P$ from focus $=$eccentricity $\times$( distance of $P$ from directrix )
$\Rightarrow \sqrt{(x-3{)}^2+(y+2{)}^2}=\frac{3}{4}\left|\frac{2x-y+3}{\sqrt{2^2+(-1{)}^2}}\right|$
Squaring both the sides
$\Rightarrow (x-3{)}^2+(y+2{)}^2=\frac{9}{16}{\left|\frac{2x-y+3}{\sqrt{5}}\right|}^2$
$\Rightarrow 80(x^2+9-6x+y^2+4+4y)=9(4x^2+y^2+9-4xy+12x-6y)$
$\Rightarrow 44x^2+71y^2+36xy-588x+374y+80\times 13-9\times 9=0$
$\Rightarrow 44x^2+36xy+71y^2-588x+374y+959=0$