Hyperbola Standard Equation

The standard equation for a hyperbola centered at the origin with its transverse axis along the x-axis is:

$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $

The vertices are given as $(\pm a, 0)$ and the foci are given as $(\pm c, 0)$.

Determining Hyperbola Parameters

From the given information:

• Vertices are $(\pm 2, 0)$, which means $a = 2$.
• Foci are $(\pm 3, 0)$, which means $c = 3$.

Now, we calculate the squares of these values:

• $a^2 = 2^2 = 4$.
• $c^2 = 3^2 = 9$.

Calculating $b^2$ and Final Equation

For a hyperbola, the relationship between $a$, $b$, and $c$ is $c^2 = a^2 + b^2$. We can use this to find $b^2$:

$ 9 = 4 + b^2 $

Solving for $b^2$:

$ b^2 = 9 - 4 = 5 $

Substitute the values of $a^2$ and $b^2$ into the standard equation:

$ \frac{x^2}{4} - \frac{y^2}{5} = 1 $

This matches Option 3.