Question

analyze this conic section to answer the questions below.
\frac{(x - 5)^2}{4}-\frac{y^2}{9}=1

a. a = 2, b = 3
b. the answer is undefined
determine the vertex or vertices for the conic section.
a. this conic section has vertices at (13,0) and (3,0).
b. this conic section has a vertex at (3,2).
c. this conic section has a vertex at (2,3).
d. this conic section has vertices at (7,0) and (3,0).
what is the value of c for this conic section? select the correct choice below and fill in any answer boxes in your choice.
a. c = (simplify your answer. type an exact answer, using radicals as needed.)
b. the answer is undefined.

Explanation:

Step1: Recall the standard form of a hyperbola

The standard form of a hyperbola with a horizontal transverse axis is \(\frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}}=1\), where \((h,k)\) is the center, \(a\) is the distance from the center to the vertices, and \(c\) is related to \(a\) and \(b\) by the formula \(c^{2}=a^{2}+b^{2}\).
For the given hyperbola \(\frac{(x - 5)^{2}}{4}-\frac{y^{2}}{9}=1\), we have \(h = 5\), \(k=0\), \(a^{2}=4\) (so \(a = 2\)), \(b^{2}=9\) (so \(b = 3\)).

Step2: Calculate \(c\)

Using the formula \(c^{2}=a^{2}+b^{2}\), substitute \(a = 2\) and \(b = 3\).

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Answer:

\(c=\sqrt{13}\)