Question

Find the value of $x$ so that the points $\left( {x, - 1} \right),\left( {2,1} \right)$ and $\left( {4,5} \right)$ are collinear.

Answer 1

Here, we will use the slope of the two points formula to find the slope of any two line segments. We will equate the slope of both the line segments by using the condition of collinearity to find the value of the variable. If two or more points lie on a line close to each other or far away from each other, then the points are said to be collinear.

Formula Used:
Slope of the line segment $m$ joining two points say $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is given by the formula: $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$

Complete step by step solution:
We are given that the points $\left( {x, - 1} \right),\left( {2,1} \right)$ and $\left( {4,5} \right)$ are collinear.
Let A $\left( {x, - 1} \right)$, B $\left( {2,1} \right)$ and C $\left( {4,5} \right)$ be the given points.
We will first find the slope of the line segment.
Now, by using the slope of the two points formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$, we get slope of the line segment A $\left( {x, - 1} \right)$and B$\left( {2,1} \right)$ as:
${m_1} = \dfrac{{1 - \left( { - 1} \right)}}{{2 - x}}$
Adding the terms in the numerator, we get
$\Rightarrow {m_1} = \dfrac{2}{{2 - x}}$ ……………………….$\left( 1 \right)$
Now, by using the slope of the two points formula, we get slope of the line segment B $\left( {2,1} \right)$ and C $\left( {4,5} \right)$ as:
${m_2} = \dfrac{{5 - 1}}{{4 - 2}}$
Subtracting the terms, we get
$\Rightarrow {m_2} = \dfrac{4}{2}$
Dividing 4 by 2, we get
$\Rightarrow {m_2} = 2$ ………………………………$\left( 2 \right)$
We know that if the given points are collinear then their slopes must be equal.
Now, by equating equation $\left( 1 \right)$ and equation $\left( 2 \right)$, we get
${m_1} = {m_2}$
$\Rightarrow \dfrac{2}{{2 - x}} = 2$
On cross multiplication, we get
$\Rightarrow 2 = 2\left( {2 - x} \right)$
Multiplying the terms, we get
$\Rightarrow 2 = 4 - 2x$
Now, by rewriting the equation, we get
$\Rightarrow 2x = 4 - 2$
$\Rightarrow 2x = 2$
Dividing both sides by 2, we get
$\Rightarrow x = 1$

Therefore, the value of $x$ is $1$.

Note:
If three or more points are given then the given points are said to be collinear if the slope of any two pairs of points is the same. This can also be found by using the area of the triangle formula. If the area of the triangle formed by three points is zero, then they are said to be collinear.
Area of the triangle using three points is given by the formula $A = \dfrac{1}{2}\left[ {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]$.
Let A $\left( {x, - 1} \right)$, B$\left( {2,1} \right)$ and C$\left( {4,5} \right)$ be the given points.
We are given that the points are collinear.
$\Rightarrow \dfrac{1}{2}\left[ {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right] = 0$
By substituting the points, we get
$\Rightarrow \dfrac{1}{2}\left[ {x\left( {1 - 5} \right) + 2\left( {5 - \left( { - 1} \right)} \right) + 4\left( {\left( { - 1} \right) - 1} \right)} \right] = 0$
Now, by simplifying the equation, we get
$\Rightarrow \dfrac{1}{2}\left[ {x\left( { - 4} \right) + 2\left( 6 \right) + 4\left( { - 2} \right)} \right] = 0$
Multiplying the terms, we get
$\Rightarrow \dfrac{1}{2}\left[ { - 4x + 12 - 8} \right] = 0$
$\Rightarrow \dfrac{1}{2}\left[ { - 4x + 4} \right] = 0$
Multiplying both sides by 2, we get
$\Rightarrow - 4x + 4 = 0$
$\Rightarrow - 4x = - 4$
Dividing both sides by $- 4$, we get
$\Rightarrow x = 1$
Therefore, the value of $x$ is $1$.