Question

If the equation $lx²+2mxy+ny²=0$ represents a pair of conjugate diameters of the hyperbola $x²/a² - y²/b² = 1$, then

• A. $la^{2}+nb^{2}=0$
• B. $la^{2}=nb^{2}$
• C. $2la^{2}=nb^{2}$
• D. None of these

Hint:

If the equation $lx=0$ represents a pair of conjugate diameters of the hyperbola $x/a - y/b = 1$, then

Answer 1

The Correct Option is B

Step 1: Concept
Two diameters $y = m_1x$ and $y = m_2x$ are conjugate for a hyperbola if $m_1m_2 = b^2/a^2$.
Step 2: Analysis
For the pair of lines $lx^2 + 2mxy + ny^2 = 0$, the product of slopes $m_1m_2 = l/n$.
Step 3: Evaluation
Equating the products: $l/n = b^2/a^2$.
Step 4: Conclusion
$la^2 = nb^2$.
Final Answer: (b)