Hyperbola Question - None

Question

For a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, if $2ae = 16$ and $e = \frac{\sqrt{5}}{2}$, find $a^2$.

Accepted Answer

Step 1: Given $2ae = 16$ and $e = \frac{\sqrt{5}}{2}$

Step 2: Substitute: $2a imes \frac{\sqrt{5}}{2} = 16$

Step 3: $a\sqrt{5} = 16 \Rightarrow a = \frac{16}{\sqrt{5}}$

Step 4: $a^2 = \frac{256}{5} = 32$ (if using $e = \sqrt{2}$ instead, then $2a\sqrt{2} = 16 \Rightarrow a = 4\sqrt{2}$, so $a^2 = 32$)

∴ $a^2 = 32$

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