Question

Find the value of $x$ for which the points $(x,–1),(2,1)\text{and}(4,5)$ are collinear.

Answer 1

Step $1:$ Slope of lines
Let the three points be $A(x,-1),B(2,1),C(4,5)$
We know that slope of a line through the points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2-y_1}{x_2-x_1}$

Slope of $AB$ through the points $(x,-1)$ and $(2,1)$
$m=\frac{1-(-1)}{2-x}$
$m=\frac{1+1}{2-x}$
$m=\frac{2}{2-x}$

Slope of line $BC$ through the points $(2,1)\&(4,5)$
$m=\frac{5-1}{4-2}$
$m=\frac{4}{2}$
$m=2$

Step $2:$ Solve for value of $x$
If three points are collinear, then their slopes must be equal.
$\therefore$ Slope of $AB$= Slope of $BC$
$\Rightarrow \frac{2}{2-x}=2$
$\Rightarrow 2=2(2-x)$
$\Rightarrow x=1$
Therefore, the value of $x\text{is}1$