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Chapter 14: Problem 1245

If the \(\mathrm{y}\) -intercept of the perpendicular bisector of the segment obtained by joining \(\mathrm{P}(1,4)\) and \(\mathrm{Q}(\mathrm{k}, 3)\) is \(-4\) then \(\mathrm{k}=\ldots \ldots\) (a) 1 (b) 2 (c) \(-2\) (d) \(-4\)

Short Answer

Expert verified
The value of k is \(-2\).

Step by step solution

01

Find the midpoint of the segment PQ

To find the midpoint, we can use the midpoint formula: \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\) Midpoint: \(\left(\frac{1+k}{2}, \frac{4+3}{2} \right) = \left(\frac{1+k}{2}, \frac{7}{2} \right)\)
02

Determine the slope of the PQ segment

To find the slope of PQ, use the slope formula: \(\frac{y_2-y_1}{x_2-x_1}\) Slope of PQ: \(\frac{3-4}{k-1} = \frac{-1}{k-1}\)
03

Find the negative reciprocal of the slope of PQ

The negative reciprocal of the PQ slope is the slope of the perpendicular bisector: Perpendicular bisector slope: \(\frac{1}{k-1}\)
04

Write the equation of the perpendicular bisector

Using the slope-point form of a linear equation (y - y1 = m(x - x1)), we can write the equation of the perpendicular bisector passing through the midpoint of PQ: \(y - \frac{7}{2} = \frac{1}{k-1}\left(x - \frac{1+k}{2}\right)\)
05

Use the given y-intercept to find the value of k

As the y-intercept is -4, we can plug (0,-4) into the equation of the perpendicular bisector: \(-4 - \frac{7}{2} = \frac{1}{k-1}\left(0 - \frac{1+k}{2}\right)\) Solving for k, we get: \(-\frac{15}{2} = -\frac{1+k}{2(k-1)}\) By cross multiplying and solving for k, we get: \(k=-2\) Hence, the correct answer is (c) -2.

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