Question

The differential equation of all lines perpendicular to the line \[ 5x + 2y + 7 = 0 \text{ is} \]

• A. \( 3dy - 2dx = 0 \)
• B. \( 2dy - 5dx = 0 \)
• C. \( 2dy - 3dx = 0 \)
• D. \( 5dy - 2dx = 0 \)

Hint:

To find the differential equation of lines perpendicular to a given line, first find the slope of the original line and take the negative reciprocal for the perpendicular slope.

Answer 1

The Correct Option is D

Step 1: Finding the slope of the given line.
The equation of the given line is \( 5x + 2y + 7 = 0 \). Rearranging this, we get: \[ 2y = -5x - 7 \quad \Rightarrow \quad y = -\frac{5}{2}x - \frac{7}{2} \] So the slope of the given line is \( m = -\frac{5}{2} \).
Step 2: Slope of the perpendicular line.
The slope of a line perpendicular to the given line is the negative reciprocal of the original slope: \[ m_{\text{perp}} = \frac{2}{5} \]
Step 3: Equation of the perpendicular line.
The general form of the equation of a line is \( y = mx + c \). For the perpendicular line, we have: \[ \frac{dy}{dx} = \frac{2}{5} \] Thus, the differential equation of all lines perpendicular to the given line is \( 5dy - 2dx = 0 \), which makes option (D) the correct answer.