Step 1: Understanding the Question:
A hyperbola is a conic section with an eccentricity greater than 1.
To find its properties, we must first convert its equation into the standard intercept form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).
From the standard form, we can identify the semi-transverse axis \( a \) and the semi-conjugate axis \( b \), which are used to calculate the latus rectum and eccentricity.
Step 2: Key Formula or Approach:
1. Standard form: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).
2. Eccentricity formula: \( e = \sqrt{1 + \frac{b^2}{a^2}} \).
3. Length of Latus Rectum (\( LR \)): \( LR = \frac{2b^2}{a} \).
Step 3: Detailed Explanation:
Step 3.1: Converting to standard form:
Divide the entire equation \( 9x^2 - 16y^2 = 144 \) by 144:
\[ \frac{9x^2}{144} - \frac{16y^2}{144} = \frac{144}{144} \]
\[ \frac{x^2}{16} - \frac{y^2}{9} = 1 \]
Comparing this with \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), we get:
\( a^2 = 16 \implies a = 4 \)
\( b^2 = 9 \implies b = 3 \)
Step 3.2: Calculating eccentricity (\( e \)):
\[ e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \frac{9}{16}} \]
\[ e = \sqrt{\frac{16 + 9}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4} \]
Step 3.3: Calculating Length of Latus Rectum (\( LR \)):
\[ LR = \frac{2b^2}{a} = \frac{2(9)}{4} \]
\[ LR = \frac{18}{4} = \frac{9}{2} \]
The pair of values is \( (LR, e) = (9/2, 5/4) \).
Step 4: Final Answer:
The length of the latus rectum is \( 9/2 \) and the eccentricity is \( 5/4 \).