Question

1. find the standard form equation for the hyperbola centered at the origin that has vertices (0, ±5) and foci (0, ±8).

Explanation:

Step1: Determine the form of the hyperbola equation

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

Step2: Find the value of \(a\)

The vertices are \((0,\pm a)\), and given vertices \((0,\pm5)\), so \(a = 5\), then \(a^{2}=25\).

Step3: Find the value of \(c\)

The foci are \((0,\pm c)\), and given foci \((0,\pm8)\), so \(c = 8\).

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

Substitute \(a = 5\) and \(c = 8\) into \(c^{2}=a^{2}+b^{2}\), we get \(b^{2}=c^{2}-a^{2}\).

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Answer:

The standard form equation of the hyperbola is \(\frac{y^{2}}{25}-\frac{x^{2}}{39}=1\)