<s>[INST] Minimize area of ellipse that passes through given point The ellipse formula is:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
$a$ and $b$ are assumed to be positive.
The point $(6, -7)$ is given as a point on the ellipse, with the area of the ellipse equal to $\pi ab$. 
How would one find the values of $a$ and $b$ that minimize the area of the ellipse?
 [/INST] Note that $$\frac{6^2}{a^2}+\frac{7^2}{b^2}=1$$
Using $AM\ge GM$,
\begin{align}
  \frac{1}{2}
  \left(
    \frac{6^2}{a^2}+\frac{7^2}{b^2}
  \right) & \ge \sqrt{\frac{6^2}{a^2} \times \frac{7^2}{b^2}} \\
  \frac{1}{2} & \ge \frac{42}{ab} \\
  \pi ab & \ge 84\pi
\end{align}
Equality holds if and only if $\dfrac{6^2}{a^2}=\dfrac{7^2}{b^2}$, therefore $\dfrac{a}{6}=\dfrac{b}{7}=\sqrt{2}$.
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