Question

analyze this conic section to answer the questions below.\
\
\\( \frac { ( x - 5 ) ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1 \\)\
\
which of the following statements is true?\
\\( \bigcirc \\) a. the transverse axis for this conic section is along the \\( y \\)-axis.\
\\( \bigcirc \\) b. this conic section has a major and a minor axis.\
\\( \bigcirc \\) c. the transverse axis for this conic section is along the \\( x \\)-axis.\
\\( \bigcirc \\) d. the directrix for this conic section is \\( y = 1 \\).\
\\( \bigcirc \\) e. the directrix for this conic section is \\( y = 9 \\).\
determine the asymptotes for the conic section.\
\\( \bigcirc \\) a. the asymptotes for this conic section are \\( y = \pm \frac { 2 } { 13 } ( x - 3 ) \\).\
\\( \bigcirc \\) b. the asymptotes for this conic section are \\( y = \pm \frac { 3 } { 2 } ( x - 5 ) \\).\
\\( \bigcirc \\) c. this conic section does not have any asymptotes.

Explanation:

Step1: Identify the conic section

The given equation \(\frac{(x - 5)^2}{4}-\frac{y^2}{9}=1\) is of the form \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\), which represents a hyperbola.

Step2: Analyze the transverse axis

For a hyperbola of the form \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\), the transverse axis is parallel to the \(x\) - axis. Here \(h = 5,k=0,a = 2,b = 3\).

Step3: Find the asymptotes

The formula for the asymptotes of a hyperbola \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\) is \(y-k=\pm\frac{b}{a}(x - h)\). Substituting \(h = 5,k = 0,a = 2,b = 3\), we get \(y=\pm\frac{3}{2}(x - 5)\)

Answer:

For the first question: C. The transverse axis for this conic section is along the \(x\) - axis.
For the second question: B. The asymptotes for this conic section are \(y=\pm\frac{3}{2}(x - 5)\)