Question

analyze this conic section to answer the questions below.\\( \frac{(x - 5)^{2}}{4}-\frac{y^{2}}{9}=1 \\)determine the focus or foci for the conic section.\\( \bigcirc \\) a. the focus for this conic section is at \\( (5-\sqrt{13},5 + \sqrt{13}) \\).\\( \bigcirc \\) b. the foci for this conic section are at \\( (5-\sqrt{13},0) \\) and \\( (5+\sqrt{13},0) \\).\\( \bigcirc \\) c. the foci for this conic section are at \\( (2,0) \\) and \\( (13,0) \\).\\( \bigcirc \\) d. the focus for this conic section is at \\( (2,3) \\).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 \\).

Explanation:

Step1: Identify the standard form of hyperbola

The given equation \(\frac{(x - 5)^2}{4}-\frac{y^2}{9}=1\) is of the form \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\) (center \((h,k)=(5,0)\), \(a^2 = 4\), \(b^2=9\)).

Step2: Calculate \(c\)

For a hyperbola \(c^2=a^2 + b^2\). Substituting \(a^2 = 4\) and \(b^2=9\), we get \(c^2=4 + 9=13\), so \(c=\sqrt{13}\).

Step3: Find foci

For a hyperbola of the form \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\), the foci are \((h\pm c,k)\). Here \(h = 5,k = 0,c=\sqrt{13}\), so foci are \((5-\sqrt{13},0)\) and \((5+\sqrt{13},0)\).

Step4: Analyze transverse axis

For the hyperbola \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\), the transverse axis is along the \(x\) - axis (since the \(x\) - term is positive).

Answer:

For the foci question: B. The foci for this conic section are at \((5-\sqrt{13},0)\) and \((5+\sqrt{13},0)\)
For the true - statement question: C. The transverse axis for this conic section is along the \(x\) - axis