Step 1: Understanding the Concept:
The given equation represents a hyperbola. We need to convert it to standard form \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) to identify the semi-axes and then calculate the eccentricity. : Key Formula or Approach:
Eccentricity of a hyperbola: \( e = \sqrt{1 + \frac{b^{2}}{a^{2}}} \). Step 2: Detailed Explanation:
Start with the equation:
\[ 9x^{2} - 16y^{2} = 144 \]
Divide both sides by 144 to get the standard form:
\[ \frac{9x^{2}}{144} - \frac{16y^{2}}{144} = 1 \]
\[ \frac{x^{2}}{16} - \frac{y^{2}}{9} = 1 \]
Comparing with standard form \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \):
\[ a^{2} = 16 \text{ and } b^{2} = 9 \]
Now calculate eccentricity \( e \):
\[ e = \sqrt{1 + \frac{9}{16}} \]
\[ e = \sqrt{\frac{16 + 9}{16}} \]
\[ e = \sqrt{\frac{25}{16}} = \frac{5}{4} \]. Step 3: Final Answer:
The eccentricity is 5/4.