Question

If the length of the latus rectum and the length of transverse axis of a hyperbola are \( 4\sqrt{3} \) and \( 2\sqrt{3} \) respectively, then the equation of the hyperbola is:

• A. \( \frac{x^2}{3} - \frac{y^2}{4} = 1 \)
• B. \( \frac{x^2}{3} - \frac{y^2}{9} = 1 \)
• C. \( \frac{x^2}{6} - \frac{y^2}{9} = 1 \)
• D. \( \frac{x^2}{6} - \frac{y^2}{3} = 1 \)
• E. \( \frac{x^2}{3} - \frac{y^2}{6} = 1 \)

Hint:

Always identify whether the given length is the "axis" ($2a$) or the "semi-axis" ($a$). Squaring the semi-axis gives you the denominator for the equation.

Answer 1

Answer: (E)

Concept: For a hyperbola in standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \):
• Length of transverse axis = \( 2a \)
• Length of latus rectum = \( \frac{2b^2}{a} \)

Step 1: Find the value of \( a \).
Length of transverse axis \( 2a = 2\sqrt{3} \). \[ a = \sqrt{3} \quad \Rightarrow \quad a^2 = 3 \]

Step 2: Find the value of \( b \).
Length of latus rectum \( \frac{2b^2}{a} = 4\sqrt{3} \). Substitute \( a = \sqrt{3} \): \[ \frac{2b^2}{\sqrt{3}} = 4\sqrt{3} \] \[ 2b^2 = 4\sqrt{3} \times \sqrt{3} \] \[ 2b^2 = 4 \times 3 = 12 \] \[ b^2 = 6 \]

Step 3: Form the equation.
Substitute \( a^2 = 3 \) and \( b^2 = 6 \) into the standard equation: \[ \frac{x^2}{3} - \frac{y^2}{6} = 1 \]