Question

Find the equation of the ellipse having its centre at $(2,-3)$ and one vertex at $(4,-3)$.

Answer 1

A] foci = (3,-3) =

distance between center & focus

CF = ae

distance between center & vertex = a

$CF=\sqrt{(2-4{)}^2+(-3+3{)}^2}=1$

$\therefore ae=1$

distance between center & vertex

$CA=\sqrt{(2-4{)}^2+(-3+3{)}^2}=2$

$\therefore a=2$

$\Rightarrow e=\frac{1}{2}$

$b^2=a^2(1-e^2)$

$b^2=4(1-\frac{1}{4})$

$b^2=4\times \frac{3}{4}=3$

So, equation is

$=\frac{(x-2{)}^2}{4}+\frac{(y+3{)}^2}{3}=1$