Question

1. find the eccentricity of the hyperbola $\frac{(x - 2)^2}{4^2}-\frac{(y - 1)^2}{2^2}=1$.

eccentricity: $e = \frac{\sqrt{a^2 + b^2}}{a}=$
$\frac{2\sqrt{5}}{4}$
$\frac{-2\sqrt{5}}{4}$
$\frac{5\sqrt{2}}{4}$
$\frac{4\sqrt{5}}{2}$

Explanation:

Step1: Identify \(a\) and \(b\)

For the hyperbola \(\frac{(x - 2)^2}{4^2}-\frac{(y - 1)^2}{2^2}=1\), we have \(a = 4\) and \(b=2\).

Step2: Calculate \(a^{2}+b^{2}\)

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Step3: Calculate the eccentricity \(e\)

Using the formula \(e=\frac{\sqrt{a^{2}+b^{2}}}{a}\), substitute \(a = 4\) and \(a^{2}+b^{2}=20\)

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Answer:

A. \(\frac{2\sqrt{5}}{4}\)