Question

For what value of x, will the point \[( - 1,x)\], \[( - 3,2)\] and \[( - 4,4)\] lie on a line?
A) \[ - 3\]
B) 3
C) \[ - 2\]
D) 2

Answer 1

Hint: To solve the given question, i.e., to find the value of x, we will take the situation that the given points lie on a line, or we can say that the given points are collinear, so we will put the area of triangle equals to zero, then in the formula after putting the values, we will get our required answer, i.e., the value of x.

Complete step by step solution:
We have been given three points \[( - 1,x)\], \[( - 3,2)\] and \[( - 4,4)\] lie on a line. We need to find the value of x. 
Since, the points are collinear points. Let us construct a figure to understand better.

Now, from the figure it is clear that the points are collinear and as the given points are collinear, then the area of the triangle formed by them should also be equal to zero.
\[ \Rightarrow \dfrac{1}{2}\left( {{x_1}({y_2} - {y_3}) + {x_2}({y_3} - {y_1}) + {x_3}({y_1} - {y_2})} \right) = 0\].
Here, \[\left( {{x_1},{y_1}} \right) = ( - 1,x)\], \[({x_2},{y_2}) = ( - 3,2)\] and \[({x_3},{y_3}) = ( - 4,4)\].
Substituting we have,
\[ \Rightarrow \dfrac{1}{2}\left( { - 1(2 - 4) - 3( - 4 - x) - 4(x - 2)} \right) = 0\]
\[ \Rightarrow \dfrac{1}{2}\left( { - 1( - 2) - 3( - 4 - x) - 4(x - 2)} \right) = 0\]
Now expanding the brackets we have,
\[ \Rightarrow \dfrac{1}{2}\left( {2 - 12 + 3x - 4x + 8} \right) = 0\]
Multiply by 2 on both sides of the equation we have,
\[ \Rightarrow 2 - 12 + 3x - 4x + 8 = 0\]
\[ \Rightarrow 10 - 12 - x = 0\]
\[ \Rightarrow - 2 - x = 0\]
Adding two on both side we have,
\[ \Rightarrow - x = 2\]
Multiply by -1 on both side we have,
\[ \Rightarrow x = - 2\]

Hence the required answer is option (C).

Note: Collinear points are those points which lie on the same line.
And the formula which we have used, for the area of the triangle. \[ \dfrac{1}{2}\left( {{x_1}({y_2} - {y_3}) + {x_2}({y_3} - {y_1}) + {x_3}({y_1} - {y_2})} \right) = 0\], this formula is used when three points are given which lie on a straight line, and we have put the area of triangle equals to zero in the solutions, because we know that three points which lie on a straight line can’t make a triangle, that’s why the area of triangle formed will also be equals to zero.