Question

The equation of the ellipse whose centre is $(2,-3)$, one of the foci is $(3,-3)$ and the corresponding vertex is $(4,-3)$.

Answer 1

equation of an ellipse is given by

$\frac{{(x-h)}^2}{a^2}+\frac{{(y-k)}^2}{b^2}=1$

given the center $(2,-3)$, foci $(3,-3)$ and vertex $(4,-3)$

now $a=4-2=2$ (from vertex to center)

$a^2=4$

$c=3-2=1$ (from foci to center)

$c^2=1$

$c^2=a^2-b^2⟹1=4-b^2⟹b^2=3$

Hence, the equation of ellipse is $\frac{{(x-2)}^2}{4}+\frac{{(y+3)}^2}{3}=1$

(Answer)