Question

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

Answer 1

The given points are collinear then the area of triangle formed by these points will be 0

$\therefore \frac{1}{2}\left[x_1\right(y_2-y_1)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]=0$

Here $(x_1,y_1)=(x,y)$

$(x_2,y_2)=(1,2)$

$(x_3,y_3)=(7,0)$

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

$⟹\frac{1}{2}[2x-y+7y-14]=0$

$⟹\frac{1}{2}[2x+6y-14]=0$

$⟹2x+6y-14=0$

$⟹x+3y-7=0$

Hence this is the required relation between $x$ and $y$.