Question

Are the points (3,-2) (5,2) (8,8) are collinear.

Answer 1

Let the points be A(3,-2) , B(5,2) and C(8,8)
We need to prove that they are collinear.
Let, B divide AC in ratio of k : 1
Then coordinates of B will be $(\frac{8k+3}{k+1},\frac{8k-2}{k+1})$
But the coordinates of B are given as B(5,2)
Comparing,
$\frac{8k+3}{k+1}=5\Rightarrow 8k+3=5k+5$
$\Rightarrow 3k=2$
$\Rightarrow k=\frac{2}{3}$
and, $\frac{8k-2}{k+1}=2\Rightarrow 8k-2=2k+2$
$\Rightarrow 6k=4$
$\Rightarrow k=\frac{4}{6}=\frac{2}{3}$
Since, the value of k is same in both x+y co-ordinates.
$\therefore$ Points A, B and C are collinear.