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Find the value of $x$ for which the points $(x,\ -1),\ ( 2,\ 1)$ and $( 4,\ 5)$ are collinear.
Given: Points $(x,\ -1),\ ( 2,\ 1)$ and $( 4,\ 5)$ are collinear.
To do: To find the value of $x$.
Solution:
Let $A( x,\ -1),\ B( 2,\ 1)$ and $C( 4,\ 5)$
As known, if points $A,\ B$ and $C$ are collinear if slope of $AB=$ slope of $BC$
Slope of $AB=\frac{1-(-1)}{2-x}=\frac{2}{2-x}$
Slope of $BC=\frac{5-1}{4-2}=\frac{4}{2}=2$
$\Rightarrow \frac{2}{2-x}=2$
$\Rightarrow \frac{1}{2-x}=1$
$\Rightarrow 2-x=1$
$\Rightarrow x=2-1$
$\therefore x=1$
Thus, given points are collinear if $x=1$.
Updated on: 10-Oct-2022
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