Question

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

Explanation:

Step1: Determine the form of the hyperbola

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

Step2: Find the value of \(a\)

The vertices of the hyperbola are \((0,\pm a)\). Given vertices \((0,\pm3)\), so \(a = 3\), and \(a^{2}=9\).

Step3: Find the value of \(c\)

The foci of the hyperbola are \((0,\pm c)\). Given foci \((0,\pm5)\), so \(c = 5\).

Step4: Calculate \(b^{2}\) using the relationship \(c^{2}=a^{2}+b^{2}\)

Substitute \(a = 3\) and \(c = 5\) into \(c^{2}=a^{2}+b^{2}\). Then \(b^{2}=c^{2}-a^{2}\).

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Answer:

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