Find the equation of the ellipse in standard form if it passes through  the points (–2,2) (3, –1)

Equation of the ellipse in standard form is $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$
Since it passes through (–2, 2) and (3, –1) then 
$\frac{4}{{\mathrm{a}}^2}+\frac{4}{{\mathrm{b}}^2}=1\to \left(1\right);\frac{9}{{\mathrm{a}}^2}+\frac{1}{{\mathrm{b}}^2}=1\to \left(2\right)$
solving (1) and (2) we get
$\begin{array}{l}\frac{1}{{\mathrm{a}}^2}=\frac{3}{32};\frac{1}{{\mathrm{b}}^2}=\frac{5}{32}\\ \therefore Required ellipse is \frac{3{\mathrm{x}}^2}{32}+\frac{5{\mathrm{y}}^2}{32}=1\\ 3{\mathrm{x}}^2+5{\mathrm{y}}^2=32\end{array}$