Question

Find the relation between $x$ and $y$ if the points $(x,y),(1,2)$ and $(7,0)$ are collinear.

Answer 1

Compute the required relation:

If given points are collinear then the area of the triangle formed by them must be zero.

Let $(x,y),(1,2)$ and $(7,0)$ be the vertices of a triangle,

Area of the triangle$=0$

We know that the area of a triangle is given as,

$\frac{1}{2}\left[x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right]=A$

$\Rightarrow$ $\frac{1}{2}\left[x\left(2-0\right)+1\left(0-y\right)+7\left(y-2\right)\right]=0$

$\Rightarrow$ $2x–y+7y–14=0$

$\Rightarrow$ $2x+6y–14=0$

$\Rightarrow$ $x+3y–7=0$

Hence, the relation between $x$ and $y$ is $x+3y–7=0$.