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 \\( \frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 64 } = 1 \\). (simplify your answer. use integers or fractions for any numbers in the expression.) choose the correct graph below. a. b. c. d.

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 the vertex \((0,6)\) is \(a\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\), with \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(0,6)\), we get \(a = 6\), so \(a^{2}=36\)

Step3: Find the value of \(c\)

The distance from the center \((0,0)\) to the focus \((0,-10)\) is \(c\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\), with \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(0,-10)\), we get \(c = 10\)

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

We know the relationship \(c^{2}=a^{2}+b^{2}\). Substituting \(a = 6\) and \(c = 10\) into the formula:

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Step5: Write the equation of the hyperbola

Substituting \(a^{2}=36\) and \(b^{2}=64\) into the standard form \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\), we get \(\frac{y^{2}}{36}-\frac{x^{2}}{64}=1\)

Step6: Analyze the graph

For the hyperbola \(\frac{y^{2}}{36}-\frac{x^{2}}{64}=1\), it opens up and down (because the \(y\) - term is positive). The vertices are at \((0,\pm6)\)

Answer:

The equation of the hyperbola is \(\frac{y^{2}}{36}-\frac{x^{2}}{64}=1\). To graph it, since it is of the form \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1\) with \(a = 6\) and \(b = 8\), it opens up - down. The correct graph (assuming the standard orientation based on the equation) is the one that opens up and down with vertices on the \(y\) - axis. If in the options, the graph that opens up - down (like a "U" and "upside - down U" along the \(y\) - axis) is the correct one. For example, if option \(C\) is the graph that opens up and down with vertices at \((0,\pm6)\) (relative to the scale), then \(C\) is the answer. But without seeing the exact details of each graph's orientation (just based on the equation \(\frac{y^{2}}{36}-\frac{x^{2}}{64}=1\)), the key is that it opens up - down.