Answer
(a) They are on the same line
(b) They are not on the same line
Work Step by Step
To determine, whether or not points are on the same line, we have to connect any two point find its slope and then connect another two point and find its slope; if the slopes are the same, then they lie on the same line.
(a) $A(1,1)$; $B(3,9)$; $C(6,21)$
$m_{AB}=\frac{9-1}{3-1}=\frac{8}{2}=4$
$m_{AC}=\frac{21-1}{6-1}\frac{20}{5}=4$
$m_{BC}=\frac{21-9}{6-3}=\frac{12}{3}=4$
$$m_{AB}=m_{AC}=m_{BC}$$
All the possible lines connecting these points have the same slope. It means, that all the points lie on the same line.
(b) $D(-1,3)$; $E(1,7)$; $F(4,15)$
$m_{DE}=\frac{7-3}{1-(-1)}=\frac{4}{2}=2$
$m_{DF}=\frac{15-3}{4-(-1)}\frac{12}{5}=2.4$
$m_{EF}=\frac{15-7}{4-1}=\frac{8}{3}$
$$m_{DE}\ne m_{DF}\ne m_{EF}$$
All the possible lines connecting these points have different slopes. It means, that all the points are located on different lines. (For a visual representation, see the image above)
