Question

Find the equation and eccentricity of the ellipse if:
(i) the centre of ellipse is same as centre of the hyperbola $4x^2-9y^2+8x-36y-68=0$
(ii) length of semi major axis is 3
(iii) length if minor axis is 2
(iv) the equation of major axis is $x=-1$

Answer 1

$4x^2-9y^2+8x-36y-68=0$

$\Rightarrow 4x^2+8x-(9y^2+36y)-68=0$

$\Rightarrow (2x{)}^2+2\times 2x\times 2+4-4-\{(3y{)}^2+2\times 3y\times 6+36-36\}-68=0$

$\Rightarrow (2x+2{)}^2-(3y+6{)}^2-36=0$

$\Rightarrow \frac{(x+1{)}^2}{3^2}-\frac{(y+2{)}^2}{2^2}=1$

Comparing with $\frac{(x-\alpha {)}^2}{a_1^2}-\frac{(y-\beta {)}^2}{b_1^2}=1$

centre of hyperbola $=(\alpha ,\beta )=(-1,-2)$

centre of ellipse $=(-1,-2)$

As we know equation of the major axis is $x=-1$, so the major axis lies parallel to y-axis

As from given question, $b=3$ & $2a=2\Rightarrow a=1$

$b^2=9$

$a^2=1$

Equation of ellipse is $\frac{(x+1{)}^2}{1}+\frac{(y+2{)}^2}{9}=1$

$e=\sqrt{1-\frac{a^2}{b^2}}=\frac{2\sqrt{2}}{3}$