Question

If the sum of slopes of lines represented by $ax^2+8xy+5y^2=0$ is twice their product, then $a =$

• A. $-4$
• B. 5
• C. $-2$
• D. $-8$

Hint:

For any pair of straight lines $Ax^2 + 2Hxy + By^2 = 0$, the denominators ($B$) of the sum and product formulas cancel out entirely when compared. You can directly write the relation as $-2H = 2A \implies A = -H$. Since $2H = 8 \implies H = 4$, we immediately get $a = -4$ in a single line!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The equation $ax^2+8xy+5y^2=0$ represents a homogeneous pair of straight lines passing through the origin. Let the slopes of these two lines be $m_1$ and $m_2$. We are given that the sum of the slopes is twice their product: $m_1 + m_2 = 2(m_1 m_2)$. We need to calculate the value of $a$.

Step 2: Key Formula or Approach:
For a standard homogeneous second-degree equation $Ax^2 + 2Hxy + By^2 = 0$, the sum and product formulas for the slopes are: $$m_1 + m_2 = -\frac{2H}{B}, \quad m_1 m_2 = \frac{A}{B}$$ Mapping our parameters: $A = a$, $2H = 8$, and $B = 5$.

Step 3: Detailed Explanation:
From our given equation, write down the slope properties: $$m_1 + m_2 = -\frac{8}{5}$$ $$m_1 m_2 = \frac{a}{5}$$ According to the problem's condition: $$m_1 + m_2 = 2(m_1 m_2)$$ Substitute the expressions we found into this condition: $$-\frac{8}{5} = 2\left(\frac{a}{5}\right)$$ Multiply both sides by 5 to eliminate denominators: $$-8 = 2a \implies a = -4$$

Step 4: Final Answer:
The value of $a$ is $-4$, which matches option (A).