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Chapter 14: Problem 1303

The nearest point on the line \(3 \mathrm{x}+4 \mathrm{y}=1\) from origin is (a) \([(7 / 25),(4 / 25)]\) (b) \([(7 / 25),(2 / 25)]\) (c) \([(3 / 25),(4 / 25)]\) (d) \([(1 / 25),(3 / 25)]\)

Short Answer

Expert verified
The nearest point on the line \(3x + 4y = 1\) from the origin is (c) \(\left(\dfrac{3}{25}, \dfrac{4}{25}\right)\).

Step by step solution

01

Find the slope of the given line

To find the slope, we rewrite the given equation (\(3x + 4y = 1\)) into the slope-intercept form. The slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. 1. Solve for \(y\): \(4y = -3x + 1\) \(y = -\dfrac{3}{4}x + \dfrac{1}{4}\) Now that we have the equation in slope-intercept form, we see that the slope, \(m\) is \(-\dfrac{3}{4}\).
02

Find the slope of the perpendicular line and its equation

Since the perpendicular line has a slope that is a negative reciprocal of the given line's slope, we find the negative reciprocal of \(-\dfrac{3}{4}\): Perpendicular line's slope = \(\dfrac{4}{3}\) Since the line passes through the origin (0,0), the equation of the perpendicular line is \(y = \dfrac{4}{3}x\).
03

Solve the system of equations

Now, we will solve for the point of intersection between the two lines: \( \begin{cases} 3x + 4y = 1\\ y = \dfrac{4}{3}x \end{cases} \) 1. Substitute \(\frac{4}{3}x\) for \(y\) in the first equation: \(3x + 4\left(\dfrac{4}{3}x\right) = 1\) 2. Solve for x: \(3x + \dfrac{16}{3}x = 1\) \(\Rightarrow (3+\dfrac{16}{3})x = 1\) \(\Rightarrow (\dfrac{25}{3})x = 1\) \(\Rightarrow x = \dfrac{3}{25}\) 3. Substitute \(x = \dfrac{3}{25}\) into the equation of the perpendicular line to find y: \(y = \dfrac{4}{3} \left(\dfrac{3}{25}\right)\) \(\Rightarrow y = \dfrac{4}{25}\) The point of intersection is \(\left(\dfrac{3}{25}, \dfrac{4}{25}\right)\).
04

Verify the result with given options

Now we check which of the given options matches our found point: (a) \(\left(\dfrac{7}{25}, \dfrac{4}{25}\right)\) (b) \(\left(\dfrac{7}{25}, \dfrac{2}{25}\right)\) (c) \(\left(\dfrac{3}{25}, \dfrac{4}{25}\right)\) (d) \(\left(\dfrac{1}{25}, \dfrac{3}{25}\right)\) The correct answer is (c) \(\left(\dfrac{3}{25}, \dfrac{4}{25}\right)\) as it matches the point we found.

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