Question

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

Answer 1

$\begin{array}{c}Inellipsedis\tan ceb/wfocusandae=\\ \sqrt{{\left(2-3\right)}^2+{\left(-2+3\right)}^2}=1\\ dis\tan ceb/wcentreandvertex\\ a=\sqrt{{\left(2-4\right)}^2+{\left(-3+3\right)}^2}=2\\ =\frac{ae}{a}=e=\frac{1}{2}\\ b^2=a^2\left(1-e^2\right)\\ b^2=4\left(1-\frac{1}{4}\right)=4\times \frac{3}{4}=3\\ equationofellipse\\ \frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1\\ \frac{{\left(x-2\right)}^2}{4}+\frac{{\left(y+3\right)}^2}{3}=1\end{array}$