Find the equation of an ellipse

Question

I have to find the equation of an ellipse which passes through the point $(3, 2)$, has center at the origin and major axis along the y-axis, i.e., is a vertical ellipse. No other info is given. I've tried a lot but it seems too hard to figure out.

Answer 1

Just put your point $(3,2)$ into $\frac {x^2}{a^2}+\frac{y^2}{b^2}=1$ and solve for $b$ to get $$ b=-\frac {2a}{\sqrt{a^2-9}}. $$ Set $a\gt 3$ and you get the family of ellipses that pass through $(3,2)$ and have center at the origin, since $$ \frac{3^2}{a^2}+\frac{2^2}{\left(-\frac {2a}{\sqrt{a^2-9}}\right)^2}=1 $$ is true for any $a$.