Question

find an equation for the hyperbola described. graph the equation. center at (0,0); focus at (0, - 10); vertex at (0,6) an equation of the hyperbola is = 1. (simplify your answer. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Determine the standard form of the hyperbola

Since the center is at \((0,0)\), and the focus \((0, - 10)\) and vertex \((0,6)\) lie on the \(y\) - axis, 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 distance from the center \((0,0)\) to a vertex \((0,6)\) is \(a\). So, \(a=\vert6 - 0\vert=6\), and \(a^{2}=36\).

Step3: Find the value of \(c\)

The distance from the center \((0,0)\) to a focus \((0,-10)\) is \(c\). So, \(c = \vert-10-0\vert=10\).

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

Using the relationship \(c^{2}=a^{2}+b^{2}\), we substitute \(a = 6\) and \(c = 10\). Then \(b^{2}=c^{2}-a^{2}\).

$$
LATEXBLOCK0
$$

Answer:

\(\frac{y^{2}}{36}-\frac{x^{2}}{64}=1\)