Question

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

Answer 1

Let ∴ P(a, b + c) = (x₁, y₁)

∴ Q(b, c + a) = (x₂, y₂)

∴ R(c, a + b) = (x₃, y₃)

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

Slope of PQ = `(y_2 - y_1)/(x_2 - x_1)`

= `(c + a - (b + c))/(b - a)`

=`(c + a - b - c)/(b - a)`

= `(a - b)/(b - a)`

= `(- (b - a))/(b - a)`

= –1

Slope of QR = `(y_3 - y_2)/(x_3 - x_2)`

= `((a + b) - (c + a))/(c - a)`

= `(a + b - c - a)/(c - b)`

= `(b - c)/(c - b)`

= `(- (c - b))/(c - b)`

= –1

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