Question

If the line \( (3x + 14y + 7) + k(5x + 7y + 6) = 0 \) is perpendicular to x-axis then the value of \( k \) is

• A. \( -2 \)
• B. \( 2 \)
• C. \( -\frac{3}{5} \)
• D. \( \frac{1}{3} \)

Hint:

A line perpendicular to x-axis is vertical, so eliminate \( y \)-term by setting its coefficient to zero.

Answer 1

The Correct Option is A

Step 1: Write the given line in standard form.
\[ (3x + 14y + 7) + k(5x + 7y + 6) = 0 \]
\[ (3 + 5k)x + (14 + 7k)y + (7 + 6k) = 0 \]

Step 2: Recall condition for perpendicular to x-axis.
A line perpendicular to x-axis is a vertical line.
So coefficient of \( y \) must be zero.

Step 3: Apply the condition.
\[ 14 + 7k = 0 \]

Step 4: Solve for \( k \).
\[ 7k = -14 \]
\[ k = -2 \]

Step 5: Verify the nature of line.
Substitute \( k = -2 \):
\[ (3 - 10)x + (14 - 14)y + (7 - 12) = 0 \]
\[ -7x - 5 = 0 \]
This is a vertical line.

Step 6: Interpretation.
Vertical lines are perpendicular to x-axis.

Step 7: Final conclusion.
\[ \boxed{-2} \]