Question

analyze this conic section to answer the questions below.
\\( \frac{(x - 5)^{2}}{4}-\frac{y^{2}}{9}=1 \\)
what type of conic section is the equation?
a. parabola: \\( (y - k)^{2}=\pm 4 a(x - h) \\) or \\( (x - h)^{2}=\pm 4 a(y - k) \\)
b. hyperbola: \\( \frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}}=1 \\) or \\( \frac{(y - k)^{2}}{a^{2}}-\frac{(x - h)^{2}}{b^{2}}=1 \\)
c. ellipse: \\( \frac{(x - h)^{2}}{a^{2}}+\frac{(y - k)^{2}}{b^{2}}=1 \\) or \\( \frac{(x - h)^{2}}{b^{2}}+\frac{(y - k)^{2}}{a^{2}}=1, a>b>0 \\)
d. circle: \\( \frac{(x - h)^{2}}{a^{2}}+\frac{(y - k)^{2}}{b^{2}}=1 \\) or \\( \frac{(x - h)^{2}}{b^{2}}+\frac{(y - k)^{2}}{a^{2}}=1, a = b \\)

Explanation:

The given equation \(\frac{(x - 5)^2}{4}-\frac{y^2}{9}=1\) is in the form \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\), which is the standard form of a hyperbola. A parabola has either \(x\) or \(y\) squared but not both. An ellipse has a sum of two fractions equal to \(1\). A circle is a special case of an ellipse with \(a = b\).

Answer:

B. Hyperbola: \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\) or \(\frac{(y - k)^2}{a^2}-\frac{(x - h)^2}{b^2}=1\)