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Question: How do you find the value of k if the line through the points \[\left( {2k - 2,27 - k} \right)\] and...

How do you find the value of k if the line through the points (2k−2,27−k)\left( {2k - 2,27 - k} \right) and (5k−5,3k+4)\left( {5k - 5,3k + 4} \right) is parallel to the line through the point (15,7)&(−9,−6)\left( {15,7} \right)\& \left( { - 9, - 6} \right) ?

Explanation

Solution

Hint : The slopes of parallel lines are equal. Note that two lines are parallel if their slopes are equal and they have different y-intercepts. In other words, perpendicular slopes are negative reciprocals of each other. Here is a quick review of the slope/intercept form of a line

Complete step-by-step answer :
Let (x1,y1)=(2k−2,27−k)\left( {{x_1},{y_1}} \right) = \left( {2k - 2,27 - k} \right)
(x2,y2)=(5k−5,3k+4)\left( {{x_2},{y_2}} \right) = \left( {5k - 5,3k + 4} \right)
(x3,y3)=(15,7)&(x4,y4)=(−9,−6)\left( {{x_3},{y_3}} \right) = \left( {15,7} \right)\& \left( {{x_4},{y_4}} \right) = \left( { - 9, - 6} \right)
Now since the lines are parallel to each other the gradient will be the same. So equating we get,
y2−y1x2−x1=y4−y3x4−x3\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \dfrac{{{y_4} - {y_3}}}{{{x_4} - {x_3}}}
Putting the values we get,
3k+4−(27−k)5k−5−(2k−2)=−6−7−9−15\dfrac{{3k + 4 - \left( {27 - k} \right)}}{{5k - 5 - \left( {2k - 2} \right)}} = \dfrac{{ - 6 - 7}}{{ - 9 - 15}}
3k+4−27+k5k−5−2k+2=−6−7−9−15\dfrac{{3k + 4 - 27 + k}}{{5k - 5 - 2k + 2}} = \dfrac{{ - 6 - 7}}{{ - 9 - 15}}
On taking k term son one side and constants on other side,
3k+k+4−275k−2k+2−5=−6−7−9−15\dfrac{{3k + k + 4 - 27}}{{5k - 2k + 2 - 5}} = \dfrac{{ - 6 - 7}}{{ - 9 - 15}}
Calculating we get,
4k−233k−3=−13−24\dfrac{{4k - 23}}{{3k - 3}} = \dfrac{{ - 13}}{{ - 24}}
Cancelling minus sign,
4k−233k−3=1324\dfrac{{4k - 23}}{{3k - 3}} = \dfrac{{13}}{{24}}
On cross multiplying we get,
24(4k−23)=13(3k−3)24\left( {4k - 23} \right) = 13\left( {3k - 3} \right)
On multiplying we get,
96k−552=39k−3996k - 552 = 39k - 39
Dividing both sides by 3 we get,
32k−184=13k−1332k - 184 = 13k - 13
Taking constants on one side and variables on other side we get,
32k−13k=184−1332k - 13k = 184 - 13
19k=17119k = 171
To find the value of k we will divide 171 by 19,
k=17119=9k = \dfrac{{171}}{{19}} = 9
So, the correct answer is “ K=9 ”.

Note : Note that the two lines are parallel and their coordinates are given so equating the gradient or slope is the only way to find the value of k.
Also if sometime we can say that the angle between the two lines is given by the formula,
tan⁡θ=∣m1−m21+m1m2∣\tan \theta = \left| {\dfrac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right| such that m1&m2{m_1}\& {m_2} are the slopes of the lines.
If the lines are parallel then m1=m2{m_1} = {m_2} .