Question

Find the value of $x$ for which the points $\left( x,-1 \right),\left( 2,1 \right)\ and\ \left( 4,5 \right)$ are collinear.

Answer 1

We will be using the concept of coordinate geometry to solve the problem. We will be using the fact that the three points are collinear if the slope of any two pairs of points is the same.

Complete step-by-step answer:

Now, we have been given three points as,

$\left( x,-1 \right),\left( 2,1 \right)\ and\ \left( 4,5 \right)$

Now, we know that if three points are collinear then the slope of any two pairs of points is the same.

Now, we know that the slope of line joining two points $\left( {{x}_{1}},{{y}_{1}} \right)\ and\ \left( {{x}_{2}},{{y}_{2}} \right)$ is $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.

Now, we have the slope of AB as,

slope of $AB =\dfrac{1+1}{2-x}$

$=\dfrac{2}{2-x}$

Also, the slope of BC is,

slope of $BC =\dfrac{5-1}{4-2}$

$=\dfrac{4}{2}$

$=2$

Now, we know that for collinear points slope of AB = slope of BC.

$\dfrac{2}{2-x}=2$

$2=4-2x$

$2x=2$

$x=1$

Therefore, the value of $x$ is 1.

Note: To solve these types of questions it is important to note that we have used the fact that if three points are collinear then the slope of any two pairs of points is the same We can also cross-check the answer by writing these points in a determinant form and equating it to Zero.