Step 1: Understanding the Concept:
We first convert the given equation into the standard form of a hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) to identify the semi-major axis \( a \) and semi-minor axis \( b \). These values are then used to calculate the length of the latus rectum and the eccentricity. Step 2: Key Formula or Approach:
1. Standard Form: Divide by the constant on the RHS.
2. Length of Latus Rectum: \( L = \frac{2b^2}{a} \).
3. Eccentricity: \( e = \sqrt{1 + \frac{b^2}{a^2}} \). Step 3: Detailed Explanation:
Divide \( 9x^2 - 16y^2 = 144 \) by 144:
\[ \frac{9x^2}{144} - \frac{16y^2}{144} = 1 \implies \frac{x^2}{16} - \frac{y^2}{9} = 1 \]
Here, \( a^2 = 16 \implies a = 4 \) and \( b^2 = 9 \implies b = 3 \).
Length of Latus Rectum:
\[ L = \frac{2b^2}{a} = \frac{2(9)}{4} = \frac{18}{4} = \frac{9}{2} \]
Eccentricity:
\[ e = \sqrt{1 + \frac{9}{16}} = \sqrt{\frac{16 + 9}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4} \] Step 4: Final Answer:
The length of the latus rectum is \( 9/2 \) and the eccentricity is \( 5/4 \).