Question

If the slope of one of the lines $ax^2 + 2hxy + by^2 = 0$ is twice that of the other, then $h^2 : ab$ is

• A. $8 : 7$
• B. $7 : 8$
• C. $9 : 8$
• D. $8 : 9$

Hint:

When one slope of the pair of lines $ax^2 + 2hxy + by^2 = 0$ is $n$ times the other ($m_1 = nm_2$), you can use the direct formula: $$4h^2n = ab(n+1)^2$$ Substituting $n = 2$ gives $4h^2(2) = ab(2+1)^2 \Rightarrow 8h^2 = 9ab \Rightarrow \frac{h^2}{ab} = \frac{9}{8}$. This shortcut delivers the answer in seconds!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The given homogeneous quadratic equation represents a pair of straight lines passing through the origin. We are given a relationship between the slopes of these two lines and need to find the ratio $h^2 : ab$.

Step 2: Key Formula or Approach:
Let the slopes of the two lines be $m_1$ and $m_2$. For the pair of straight lines $ax^2 + 2hxy + by^2 = 0$, the standard relations for the sum and product of the slopes are: $$m_1 + m_2 = -\frac{2h}{b}$$ $$m_1 m_2 = \frac{a}{b}$$ We are given the condition $m_1 = 2m_2$. We will substitute this condition into the slope relations to eliminate the slope variables and find the ratio.

Step 3: Detailed Explanation:
Given the condition $m_1 = 2m_2$, substitute this into the sum of slopes formula: $$2m_2 + m_2 = -\frac{2h}{b}$$ $$3m_2 = -\frac{2h}{b} \Rightarrow m_2 = -\frac{2h}{3b}$$ Squaring both sides gives: $$m_2^2 = \frac{4h^2}{9b^2}$$ Next, substitute the condition $m_1 = 2m_2$ into the product of slopes formula: $$(2m_2) \cdot m_2 = \frac{a}{b}$$ $$2m_2^2 = \frac{a}{b} \Rightarrow m_2^2 = \frac{a}{2b}$$ Equating the two expressions for $m_2^2$: $$\frac{4h^2}{9b^2} = \frac{a}{2b}$$ Since $b \neq 0$, we can cancel one factor of $b$ from both denominators: $$\frac{4h^2}{9b} = \frac{a}{2}$$ Cross-multiplying to isolate the variables: $$8h^2 = 9ab$$ $$\frac{h^2}{ab} = \frac{9}{8}$$ Thus, the ratio $h^2 : ab$ is $9 : 8$.

Step 4: Final Answer:
The ratio is equal to $9 : 8$, which corresponds to option (C).