Question

test (cs1): conic sections which of the following equations will produce the graph shown below? a. $\frac{y^{2}}{2}-\frac{x^{2}}{4}=1$ b. $x^{2}-\frac{y^{2}}{4}=1$ c. $\frac{y^{2}}{9}-\frac{x^{2}}{4}=1$ d. $y^{2}-\frac{x^{2}}{9}=1$

Explanation:

Step1: Recall hyperbola equation form

The general form of a vertical - oriented hyperbola is $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1$, and its vertices are at $(0,\pm a)$.

Step2: Identify vertices from the graph

From the graph, the vertices of the hyperbola are at $(0, 3)$ and $(0, - 3)$. So $a = 3$, and $a^{2}=9$.

Step3: Match the equation

Among the given options, the equation $\frac{y^{2}}{9}-\frac{x^{2}}{4}=1$ has $a^{2}=9$ which is consistent with the vertices of the graph.

Answer:

C. $\frac{y^{2}}{9}-\frac{x^{2}}{4}=1$