Question

find an equation for the hyperbola described. graph the equation.
center at (0,0), focus at (0,3), vertex at (0,1)
an equation of the hyperbola is (square = 1).
(use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Determine the form of the hyperbola equation

Since the center is at \((0,0)\), and the focus \((0,3)\) and vertex \((0,1)\) have the same \(x -\)coordinate, the hyperbola has a vertical transverse axis. The standard form of the equation is \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1\).

Step2: Find the value of \(a\)

The distance from the center \((0,0)\) to a vertex \((0,1)\) is \(a\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\), for \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(0,1)\), we have \(a=\vert1 - 0\vert=1\), so \(a^{2}=1\).

Step3: Find the value of \(c\)

The distance from the center \((0,0)\) to a focus \((0,3)\) is \(c\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\), for \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(0,3)\), we have \(c=\vert3 - 0\vert = 3\).

Step4: Find the value of \(b^{2}\)

We know the relationship \(c^{2}=a^{2}+b^{2}\). Substituting \(a = 1\) and \(c = 3\) into the formula, we get \(b^{2}=c^{2}-a^{2}\). Then \(b^{2}=3^{2}-1^{2}=9 - 1=8\).

Answer:

\(\frac{y^{2}}{1}-\frac{x^{2}}{8}=1\)