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A normal to the hyperbola $\dfrac{x^2}{4}-\dfrac{y^2}{1}=1$ has equal intercepts on the positive $x$-axis and $y$-axis. If this normal also touches the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, what is the value of $a^2+b^2$?

1. $5$
2. $25$
3. $16$
4. $25/3$

Correct answer: $25/3$

Solution

The normal to the hyperbola with equal positive intercepts can be written in a simple intercept form. Using the condition that this line is tangent to the ellipse leads to a relation between $a$ and $b$, and the required sum is $25/3$.

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