Question

The equation of a straight line passes through the point (4,-5) and is perpendicular to the straight line 3x + 4y + 5 = 0.

• A. 4x - 3y - 31 = 0
• B. 4x - 3y - 1 = 0
• C. 4x - 3y + 1 = 0
• D. 4x - 3y + 31 = 0

Hint:

A quick shortcut: a line perpendicular to \(Ax + By + C = 0\) will have the form \(Bx - Ay + K = 0\). For the line \(3x + 4y + 5 = 0\), a perpendicular line is \(4x - 3y + K = 0\). Substitute the point (4, -5) to find K: \(4(4) - 3(-5) + K = 0 \Rightarrow 16 + 15 + K = 0 \Rightarrow 31 + K = 0 \Rightarrow K = -31\). So the equation is \(4x - 3y - 31 = 0\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to find the equation of a line given a point it passes through and a line it is perpendicular to. The key relationship is between the slopes of two perpendicular lines.
Step 2: Key Formula or Approach:
1. Find the slope (\(m_1\)) of the given line. For a line \(Ax + By + C = 0\), the slope is \(m = -A/B\).
2. Find the slope (\(m_2\)) of the perpendicular line. The relationship is \(m_2 = -1/m_1\).
3. Use the point-slope form of a line equation, \(y - y_1 = m_2(x - x_1)\), to find the equation of the required line.
4. Convert the equation to the general form \(Ax + By + C = 0\).
Step 3: Detailed Explanation:
1. Find the slope of the given line.
The given line is \(3x + 4y + 5 = 0\).
Its slope, \(m_1\), is \(-\frac{A}{B} = -\frac{3}{4}\).
2. Find the slope of the perpendicular line.
The slope of our required line, \(m_2\), must be the negative reciprocal of \(m_1\). \[ m_2 = -\frac{1}{m_1} = -\frac{1}{(-3/4)} = \frac{4}{3} \] 3. Use the point-slope form.
The required line passes through the point \((x_1, y_1) = (4, -5)\) and has a slope \(m_2 = 4/3\). \[ y - y_1 = m_2(x - x_1) \] \[ y - (-5) = \frac{4}{3}(x - 4) \] \[ y + 5 = \frac{4}{3}(x - 4) \] 4. Convert to general form.
Multiply both sides by 3 to eliminate the fraction: \[ 3(y + 5) = 4(x - 4) \] \[ 3y + 15 = 4x - 16 \] Rearrange the terms to match the form \(Ax + By + C = 0\): \[ 0 = 4x - 3y - 16 - 15 \] \[ 4x - 3y - 31 = 0 \] Step 4: Final Answer:
The equation of the line is \(4x - 3y - 31 = 0\). This matches option (1).