Question

analyze this conic section to answer the questions below.
\\( \frac { ( x - 5 ) ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1 \\)
a. (5,0) (type an ordered pair.)
b. the answer is undefined.
what are the values of a and b for this conic section? select the correct choice below and fill in any answer boxes in your choice.
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)

Explanation:

Step1: Recall the standard form of hyperbola

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

Step2: Identify \(a\) and \(b\)

For the given equation \(\frac{(x - 5)^{2}}{4}-\frac{y^{2}}{9}=1\), we have \(a^{2}=4\) and \(b^{2}=9\).
Taking square roots, \(a = 2\) and \(b = 3\).

Step3: Find the center \((h,k)\)

Comparing with the standard form \(\frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}} = 1\), the center \((h,k)=(5,0)\).

Step4: Find the vertices

The vertices of a hyperbola \(\frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}} = 1\) are \((h\pm a,k)\).
Substituting \(h = 5\), \(a = 2\), \(k = 0\), we get \((5 + 2,0)=(7,0)\) and \((5-2,0)=(3,0)\)

Answer:

For the values of \(a\) and \(b\): \(a = 2\), \(b = 3\).
For the vertices: This conic section has vertices at \((7,0)\) and \((3,0)\) (Option D).