Question

The length of the latus rectum and eccentricity of the Hyperbola $9x^{2} - 16y^{2} = 144$ are}

• A. $(9/2, 5/2)$
• B. $(9/2, 5/4)$
• C. $(9, 5/4)$
• D. $(9, 5/2)$

Hint:

For hyperbola $e > 1$. Use $b^2 = a^2(e^2 - 1)$ to remember the relationship.

Answer 1

The Correct Option is B

Step 1: Concept
Convert the equation to standard form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ to find $a^2$ and $b^2$.

Step 2: Meaning
Dividing by 144: $\frac{x^2}{16} - \frac{y^2}{9} = 1$. Here $a^2 = 16 (a=4)$ and $b^2 = 9 (b=3)$.

Step 3: Analysis
Length of Latus Rectum = $2b^2/a = 2(9)/4 = 18/4 = 9/2$. Eccentricity $e = \sqrt{1 + b^2/a^2} = \sqrt{1 + 9/16} = \sqrt{25/16} = 5/4$.

Step 4: Conclusion
The pair is $(9/2, 5/4)$.
Final Answer: (B)