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Chapter 1: Problem 47

Find an equation of the line that passes through the point \((2,4)\) and is perpendicular to the line \(3 x+4 y-22=0\).

Short Answer

Expert verified
The equation of the line that passes through the point \((2,4)\) and is perpendicular to the line \(3x + 4y - 22 = 0\) is \(\boxed{4x - 3y + 4 = 0}\).

Step by step solution

01

Find the slope of the given line

Rewrite the given equation, \(3x + 4y - 22 = 0\), in the slope-intercept form \(y = mx + b\). Here, \(m\) is the slope and \(b\) is the y-intercept. \[ 4y = -3x + 22 \] \[ y = \frac{-3}{4}x + \frac{22}{4} \] So the slope of the given line \(m_1 = \frac{-3}{4}\).
02

Find the slope of the required line

If two lines are perpendicular, the slopes of these lines are negative reciprocals of each other i.e., \(m_1 \cdot m_2 = -1\). In our case, \(m_1 = \frac{-3}{4}\). Solve for \(m_2\): \[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{3}{4}} = \frac{4}{3} \]
03

Use the point-slope form to find the equation of the required line

Since we have the slope \(m_2 = \frac{4}{3}\) and the point \((2,4)\) through which the required line passes, we can use the point-slope form: \(y - y_1 = m_2 (x - x_1)\). Plug in the values: \[ y - 4 = \frac{4}{3}(x - 2) \]
04

Simplify and rewrite in standard form

Expand and then rearrange the equation to get it in the standard form: \begin{align*} y - 4 &= \frac{4}{3}(x - 2) \\ y - 4 &= \frac{4}{3}x - \frac{8}{3} \\ 3(y - 4) &= 4(x - 2) \\ 3y - 12 &= 4x - 8 \\ 4x - 3y + 4 &= 0 \end{align*} So the equation of the line that passes through the point \((2,4)\) and is perpendicular to the given line is \(\boxed{4x - 3y + 4 = 0}\).

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