Question

Let $a \ne 1$ be a non-zero real number. If the lines $2x+ay=1$ and $x+2y=1$ are perpendicular, then the value of $a$ is equal to ________.

• A. 1
• B. -2
• C. 2
• D. -1
• E. $-\frac{1}{2}$

Hint:

For perpendicular lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, $a_1a_2 + b_1b_2 = 0$.

Answer 1

The Correct Option is D

Step 1: Concept
Two lines are perpendicular if the product of their slopes ($m_1 \cdot m_2$) is $-1$.

Step 2: Meaning
Slope of first line ($2x+ay=1$) is $m_1 = -\frac{2}{a}$. Slope of second line ($x+2y=1$) is $m_2 = -\frac{1}{2}$.

Step 3: Analysis
Setting $m_1 \cdot m_2 = -1$:
$\left(-\frac{2}{a}\right) \cdot \left(-\frac{1}{2}\right) = -1 \implies \frac{1}{a} = -1$.

Step 4: Conclusion
Hence, $a = -1$. Final Answer: (D)