Question

Equation of the ellipse with focus $(3,-2)$, eccentricity $\frac{1}{4}$ and directrix $2x-y+3=0$ is :

• A. $76x^2+79y^2-4xy-492x-326y-1031=0$
• B. $76x^2+79y^2+4xy-492x+326y+1031=0$
• C. $44x^2+36xy+71y^2-125x-274y+659=0$
• D. $44x^2+36xy+71y^2-135x-47xy+859=0$

Answer 1

Answer: (B)

$76x^2+79y^2+4xy-492x+326y+1031=0$

Focus $S(3,-2)$
$e=\frac{1}{4}$
Equation of directrix is $2x-y+3=0$
Let $P(x,y)$ be a point on the ellipse.
Since, for an ellipse
$P{S}^2=e^{2}P{M}^2$
$\Rightarrow (x-3{)}^2+(y+2{)}^2=\frac{1}{16}\frac{(2x-y+3{)}^2}{5}$
$\Rightarrow 76x^2+79y^2+4xy-492x+326y+1031=0$