Question

The latus rectum of the hyperbola \(\frac{(3x - 7)^2}{9} - \frac{(4y + 3)^2}{8} = 1\) is

• A. \(\frac{1}{2}\)
• B. \(\frac{16}{3}\)
• C. 4
• D. \(\frac{4}{3}\)
• E. 1

Hint:

Be careful with the denominators. In the term \(\frac{(4y+3)^2}{8}\), the 16 from \((4)^2\) moves to the denominator as \(8/16 = 1/2\). Don't just look at the visible denominator '8'.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The equation is given in a non-standard form where the coefficients of the squared terms are not 1. We must simplify it to the form \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\).

Step 2: Key Formula or Approach:
Length of latus rectum = \(\frac{2b^2}{a}\).

Step 3: Detailed Explanation:
Rewrite the terms:
\[ \frac{9(x - 7/3)^2}{9} - \frac{16(y + 3/4)^2}{8} = 1 \]
Simplify the fractions:
\[ (x - 7/3)^2 - \frac{(y + 3/4)^2}{1/2} = 1 \]
Now, compare with \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\):
\(a^2 = 1 \implies a = 1\)
\(b^2 = 1/2\)
Calculate latus rectum:
\[ \text{L.R.} = \frac{2b^2}{a} = \frac{2(1/2)}{1} = 1 \]

Step 4: Final Answer:
The latus rectum of the hyperbola is 1.