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Chapter 14: Problem 1301

One side of the rectangle lies along the line \(4 \mathrm{x}+7 \mathrm{y}+5=0\). Two of its vertices are \((-3,1)\) and \((1,1)\). Then the equations of other side is (a) \(7 x-4 y+25=0\) (b) \(4 x+7 y=11\) (c) \(7 x-4 y-3=0\) (d) All of these

Short Answer

Expert verified
The equation of the other side of the rectangle is \(7x - 4y - 3 = 0\).

Step by step solution

01

Find the slope of the given line

We need to find the slope of the line with the equation \(4x+7y+5=0\). To do this, we will first solve for y. As, \[ 7y = -4x - 5 \Rightarrow y = -\frac{4}{7}x - \frac{5}{7} \] The slope of the given line is \(-\frac{4}{7}\).
02

Find the midpoint of the line segment

We are given the two vertices of the rectangle: \((-3,1)\) and \((1,1)\). We can find the midpoint of the line segment formed by these points using the midpoint formula: \(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\) Using the vertices \((-3,1)\) and \((1,1)\) , the midpoint M is calculated as: \(M = \left(\frac{-3+1}{2}, \frac{1+1}{2}\right) = (-1, 1)\)
03

Find the equation of the line parallel to the given line and passing through the midpoint

We know the slope of the given line and the midpoint of the line segment. Now we can find the equation of the line parallel to the given line and passing through the midpoint. The slope of the line parallel to the given line will also be \(-\frac{4}{7}\). Let's use the point-slope form of the equation: \( y- y_1 = m(x - x_1)\) With slope \(m = -\frac{4}{7}\) and point \((-1, 1)\), plug in the values and solve for y: \[ y - 1 = -\frac{4}{7}(x + 1) \Rightarrow 7y - 7 = -4x - 4 \Rightarrow 7y - 4x = 3 \] Thus, the equation of the line parallel to the given line and passing through the midpoint is \(7y - 4x = 3\).
04

Identify the correct equation

Based on our calculations, the equation of the other side of the rectangle should be \(7y - 4x = 3\). Comparing this with the given options, we find that option (c) is the correct one. The equation of the other side of the rectangle is: \(7x - 4y - 3 = 0\)

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Most popular questions from this chapter

If \(\mathrm{P}(1,2), \mathrm{Q}(4,6), \mathrm{R}(5,7)\) and \(\mathrm{S}(\mathrm{a}, \mathrm{b})\) are the vertices of a parallelogram PQRS then (a) \(a=2, b=4\) (b) \(a=3, b=4\) (c) \(\mathrm{a}=2, \mathrm{~b}=3\) (d) \(a=2, b=5\)

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