Question

(1, 6), (3, -2) and (-2, K) are collinear points. What is the value of K?
(a) -6
(b) 2
(c) 8
(d) 10
(e) 18

Answer 1

Assume the given points as A(1, 6) ; B(3, -2); and C(-2, K). Find the slope of line segment AB and BC by using the formula: $slope=\dfrac{\Delta y}{\Delta x}$. Then equate the two slopes equal to each other and simplify the expression to get the answer.

Complete step-by-step answer :
In geometry, collinear points are the set of points which lie on a single straight line.
Now, let us come to the question. We have been given that, (1, 6), (3, -2) and (-2, K) are collinear points. Let us assume these points as A, B and C respectively. Since, all the three points lie on the same line, their slopes must be equal. Therefore,
We know that, slope of line segment $=\dfrac{\Delta y}{\Delta x}$
Therefore, slope of line segment AB $=\dfrac{6-(-2)}{1-3}=\dfrac{8}{-2}=-4$
And, slope of line segment BC $=\dfrac{K-(-2)}{-2-3}=\dfrac{K+2}{-5}$
On equating these slopes equal to each other, we get,
$\dfrac{K+2}{-5}=-4$
By cross-multiplication, we get,
$\begin{aligned} & K+2=20 \\\ & \Rightarrow K=18 \\\ \end{aligned}$
Hence, option (e) is the correct answer.

Note :Here, in this question we have equated the slope of line segment AB and BC. One can also equate the slope of line segment AB and AC or the slope of line segment AC and BC. The value of K will remain the same. One important thing is that we cannot use distance formulas because we don’t know properly which point is in the middle.