Question
Let $${P_r}\left( {{x_r},\,{y_r},\,{z_r}} \right);\,r = 1,\,2,\,3;$$ be three points where $${x_1},\,{x_2},\,{x_3};\,{y_1},\,{y_2},\,{y_3}$$ and $${z_1},\,{z_2},\,{z_3}$$ are each in GP with the same common ratio. Then $${P_1},\,{P_2},\,{P_3}$$ are :
A.
coplanar points
B.
collinear points
C.
vertices of an equilateral triangle
D.
none of these
Answer :
collinear points
Solution :
$${P_1} = \left( {{x_1},\,r{x_1},\,{r^2}{x_1}} \right),\,{P_2} = \left( {{y_1},\,r{y_1},\,{r^2}{y_1}} \right),\,{P_3} = \left( {{z_1},\,r{z_1},\,{r^2}{z_1}} \right)$$
Direction ratios of $${P_1}{P_2}$$ are $${y_1} - {x_1},\,r\left( {{y_1} - {x_1}} \right),\,{r^2}\left( {{y_1} - {x_1}} \right).$$
Direction ratios of $${P_2}{P_3}$$ are $${z_1} - {y_1},\,r\left( {{z_1} - {y_1}} \right),\,{r^2}\left( {{z_1} - {y_1}} \right).$$
$${\text{As }}\frac{{{y_1} - {x_1}}}{{{z_1} - {y_1}}} = \frac{{r\left( {{y_1} - {x_1}} \right)}}{{r\left( {{z_1} - {y_1}} \right)}} = \frac{{{r^2}\left( {{y_1} - {x_1}} \right)}}{{{r^2}\left( {{z_1} - {y_1}} \right)}},\,{P_1}{P_2}||{P_2}{P_3}$$
So $${P_1},\,{P_2},\,{P_3}$$ are collinear.