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

Given:

Points $A( x,\ y),\ B( 1,\ 2)$ and $C( 7,\ 0)$ are collinear.

To do: 

We have to find the relation between $x$ and $y$. 

Solution:

Let points $A( x,\ y),\ B( 1,\ 2)$ and $C( 7,\ 0)$ are collinear.

Since the given points are collinear, therefore the area of the triangle formed by them must be zero.

We know that,

Area of triangle $=\frac{1}{2}[x_1( y_2-y_1)+x_2( y_3-y_1)+x_3( y_1-y_3)]$

Here, $x_{1}=x, y_{1}=y, x_{2}=1, y_{2}=2, x_{3}=7$ and $y_{3}=0$

Therefore,

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

$\Rightarrow \frac{1}{2}[2 x-y+7 y-14]=0$

$\Rightarrow \frac{1}{2}[2 x+6 y-14]=0$

$\Rightarrow x+3 y-7=0$

Therefore, the relation between $x$ and $y$ is $x+3y-7=0$.

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Updated on: 10-Oct-2022

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