Question

If the lines 7x+6y+9=0 and ax+14y+8=0 are perpendicular to each other, find the value of

• A. 12
• B. -12
• C. 6
• D. -6
• E. None of these

Answer 1

The Correct Option is B

Step 1: Understand the condition for perpendicular lines.
Two lines are perpendicular if the product of their slopes is -1.
The general form of a line is \( Ax + By + C = 0 \). The slope of the line is given by \( m = -\frac{A}{B} \).
Given the two lines: - Line 1: \( 7x + 6y + 9 = 0 \) - Line 2: \( ax + 14y + 8 = 0 \)

Step 2: Find the slopes of the given lines.
The slope of Line 1 is \( m_1 = -\frac{7}{6} \) (from \( 7x + 6y + 9 = 0 \)).
The slope of Line 2 is \( m_2 = -\frac{a}{14} \) (from \( ax + 14y + 8 = 0 \)).

Step 3: Use the condition for perpendicular lines.
The lines are perpendicular, so their slopes satisfy the condition:
\( m_1 \times m_2 = -1 \)
Substituting the values of \( m_1 \) and \( m_2 \):
\( -\frac{7}{6} \times -\frac{a}{14} = -1 \)
\( \frac{7a}{84} = -1 \)
\( 7a = -84 \)
\( a = -12 \)

Step 4: Conclusion.
The value of \( a \) is -12.

Final Answer:
The correct option is (B): -12.