Question

find the standard form equation for the hyperbola centered at the origin that has vertices $(0,\pm3)$ and foci $(0,\pm5)$. standard form

Explanation:

Step1: Determine the form of the hyperbola

Since the vertices and foci are on the \(y -\)axis, the standard form of the hyperbola is \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1\).
The vertices are \((0,\pm a)\), so \(a = 3\).
The foci are \((0,\pm c)\), so \(c = 5\).

Step2: Find \(b^{2}\)

Use the relationship \(c^{2}=a^{2}+b^{2}\).
Substitute \(a = 3\) and \(c = 5\) into the formula:
\(b^{2}=c^{2}-a^{2}\)
\(b^{2}=5^{2}-3^{2}\)
\(b^{2}=25 - 9\)
\(b^{2}=16\)

Step3: Write the equation

Substitute \(a^{2}=9\) and \(b^{2}=16\) into the standard form \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\).
The equation is \(\frac{y^{2}}{9}-\frac{x^{2}}{16}=1\)

Answer:

\(\frac{y^{2}}{9}-\frac{x^{2}}{16}=1\)