Question

If the equation of the ellipse is $3x^2+2y^2+6x-8y+5=0$, then which of the following is/are true?

• A. $e=\frac{1}{\sqrt{3}}$
• B. Center is $(-1,2)$
• C. Foci are $(-1,1)$ and $(-1,3)$
• D. Directrices are $y=2\pm \sqrt{3}$

Answer 1

Answer: (A), (B), (C)

$e=\frac{1}{\sqrt{3}}$
Center is $(-1,2)$
Foci are $(-1,1)$ and $(-1,3)$

$3x^2+2y^2+6x-8y+5=0$
$\Rightarrow \frac{(x+1{)}^2}{2}+\frac{(y-2{)}^2}{3}=1$
Therefore, centre is $(-1,2)$ and ellipse is vertical
$(\because b>a)$
$a^2=2,b^2=3$
Now $2=3(1-e^2)$
$\Rightarrow e=\frac{1}{\sqrt{3}}$
Foci are $(-1,2\pm be)$ and $(-1,2\pm 1)$.
Hence, the foci are $(-1,3)$ and $(-1,1)$
The equations of the directrices are $y=2\pm \frac{b}{e}\Rightarrow y=5$ and $y=-1$