Question

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

Answer 1

$C=(2,-3),F=(3,-3),vertexA=(4,-3)$ distance between centre and because is CF, ac distance between centre and vertex is $CA=a$

$\therefore CF=ac$

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

$\Rightarrow ac=1$

$\therefore CA=a$

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

$\Rightarrow a=2$

$\therefore C=1/2$

$\therefore b^2=a^2(1-C^2)$

$=4(1-\frac{1}{4})$

$=4\left(\frac{3}{4}\right)$

$b^2=3$

Since the major axis is along $\times$ axis

$\therefore$ the eq. of the ellipse is

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