Question

the graph of a hyperbola is given. match the graph to its equation below.

choose the correct answer below.

$$\frac { x ^ { 2 } } { 25 } - y ^ { 2 } = 1$$

$$\frac { y ^ { 2 } } { 25 } - x ^ { 2 } = 1$$

$$y ^ { 2 } - \frac { x ^ { 2 } } { 25 } = 1$$

$$x ^ { 2 } - \frac { y ^ { 2 } } { 25 } = 1$$

Explanation:

Step1: Recall the standard form of hyperbola

The standard form of a hyperbola is $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1$ (opens up and down) and $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ (opens left and right).

Step2: Analyze the given graph

The graph opens up and down. So the equation should be of the form $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1$.

Step3: Find the value of \(a\)

From the graph, the vertices are at \((0,5)\) and \((0, - 5)\). For the hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$, the vertices are \((0,\pm a)\). So \(a = 5\) and \(a^{2}=25\).

Answer:

\(\frac{y^{2}}{25}-x^{2}=1\)