Question

Show that the points $P(a,b+c),Q(b,c+a)$ and $R(c,a+b)$ are collinear.

Answer 1

Given: $P(a,b+c),Q(b,c+a)$ and $R(c,a+b)$

We know that, the points P, Q and R will be collinear if slope of PQ and QR is the same.

$\because \text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Slope of $PQ=\frac{c+a-b-c}{b-a}=\frac{a-b}{b-a}=-1$

Slope of $QR=\frac{a+b-c-a}{c-b}=\frac{b-c}{c-b}=-1$

$\Rightarrow$ Slope of PQ $=$ slope of QR

So, the points P, Q and R are collinear.

Hence proved.