Question

The slopes of the lines represented by $6x^2 + 2hxy + y^2 = 0$ are in the ratio 2 : 3, then $h = \dots$

• A. $\pm 7/2$
• B. $\pm 1/2$
• C. $\pm 5/2$
• D. $\pm 2/5$

Hint:

There is a direct shortcut formula! If the slopes of $ax^2 + 2hxy + by^2 = 0$ are in the ratio $m:n$, then:
$\frac{(2h)^2}{ab} = \frac{(m+n)^2}{mn}$.
Here: $\frac{4h^2}{(6)(1)} = \frac{(2+3)^2}{(2)(3)} \implies \frac{4h^2}{6} = \frac{25}{6} \implies 4h^2 = 25 \implies h = \pm 5/2$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For a pair of straight lines $ax^2 + 2hxy + by^2 = 0$, the sum of slopes $m_1 + m_2 = -2h/b$ and the product of slopes $m_1m_2 = a/b$.

Step 2: Formula Application:
Here $a = 6, b = 1, 2h = 2h$. $m_1 + m_2 = -2h$ and $m_1m_2 = 6$. Let $m_1 = 2k$ and $m_2 = 3k$.

Step 3: Explanation:
Product: $(2k)(3k) = 6 \implies 6k^2 = 6 \implies k^2 = 1 \implies k = \pm 1$. Sum: $2k + 3k = -2h \implies 5k = -2h$. If $k = 1, h = -5/2$. If $k = -1, h = 5/2$. So $h = \pm 5/2$.

Step 4: Final Answer:
The value of $h$ is $\pm 5/2$.