Question

analyze this conic section to answer the questions below
\\( \frac{(x - 5)^{2}}{4}-\frac{y^{2}}{9}=1 \\)
determine the asymptotes for the conic section
a. the asymptotes for this conic section are \\( y=\pm \frac{2}{13}(x - 3) \\).
b. the asymptotes for this conic section are \\( y=\pm \frac{3}{2}(x - 5) \\)
c. this conic section does not have any asymptotes
choose the correct graph of the conic section below.

Explanation:

Step1: Recall the standard form of a hyperbola

The standard form of a hyperbola \(\frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}}=1\) has center \((h,k)\), and its asymptotes are given by the equations \(y - k=\pm\frac{b}{a}(x - h)\).
For the given hyperbola \(\frac{(x - 5)^{2}}{4}-\frac{y^{2}}{9}=1\), we have \(h = 5\), \(k = 0\), \(a^{2}=4\) (so \(a = 2\)), and \(b^{2}=9\) (so \(b = 3\)).

Step2: Find the equations of the asymptotes

Substitute \(h = 5\), \(k = 0\), \(a = 2\), and \(b = 3\) into the asymptote formula \(y - k=\pm\frac{b}{a}(x - h)\).
We get \(y-0=\pm\frac{3}{2}(x - 5)\), which simplifies to \(y=\pm\frac{3}{2}(x - 5)\).

Answer:

B. The asymptotes for this conic section are \(y=\pm\frac{3}{2}(x - 5)\)