Answer
a) collinear
b) not collinear
Work Step by Step
We will note the given points $A, B,C$. We will calculate the slopes of the lines $AB$ and $BC$. If they are equal, the lines $AB$ and $BC$ are parallel, but because the point $B$ belongs to both of them, it means they are coincident, so the given points are collinear.
a) We are given the points $A(1,1), B(3,9),C(6,21)$.
Determine the slope of the line passing through $A(1,1)$ and $B(3,9)$:
$$m_{AB}=\dfrac{y_B-y_A}{x_B-x_A}=\dfrac{9-1}{3-1}=4.$$
Determine the slope of the line passing through $B(3,9)$ and $C(6,21)$:
$$m_{BC}=\dfrac{y_C-y_B}{x_C-x_B}=\dfrac{21-9}{6-3}=4.$$
As $m_{AB}=m_{BC}$, it means that $AB\parallel BC$, so $A,B,C$ are collinear.
b) We are given the points $A(-1,3), B(1,7),C(4,15)$.
Determine the slope of the line passing through $A(-1,3)$ and $B(1,7)$:
$$m_{AB}=\dfrac{y_B-y_A}{x_B-x_A}=\dfrac{7-3}{1-(-1)}=2.$$
Determine the slope of the line passing through $B(1,7)$ and $C(4,15)$:
$$m_{BC}=\dfrac{y_C-y_B}{x_C-x_B}=\dfrac{15-7}{4-1}=\dfrac{8}{3}.$$
As $m_{AB}\not=m_{BC}$, it means that $AB\not\parallel BC$, so $A,B,C$ are not collinear.
