Question
If $${x_1},{x_2},\,{x_3}$$ as well as $${y_1},{y_2},\,{y_3},$$ are in G.P. with the same common ratio, then the points $$\left( {{x_1},\,{y_1}} \right),\,\left( {{x_2},\,{y_2}} \right)$$ and $$\left( {{x_3},\,{y_3}} \right).$$
A.
lie on a straight line
B.
lie on an ellipse
C.
lie on a circle
D.
are vertices of a triangle
Answer :
lie on a straight line
Solution :
\[\begin{array}{l}
{x_2} = {x_1}r,\,{x_3} = {x_1}{r^2}{\rm{ \,and \,so\, }}{y_2} = {y_1}r,\,{y_3} = {y_1}{r^2}\\
\Delta = \left| \begin{array}{l}
{x_1}\,\,\,{y_1}\,\,\,1\\
{x_2}\,\,\,{y_2}\,\,\,1\\
{x_3}\,\,\,{y_3}\,\,\,1
\end{array} \right| = r.\,{r^2}\left| \begin{array}{l}
{x_1}\,\,\,{y_1}\,\,\,1\\
{x_1}\,\,\,{y_1}\,\,\,1\\
{x_1}\,\,\,{y_1}\,\,\,1
\end{array} \right| = 0
\end{array}\]
Hence the points lie on a line, i.e., they are collinear.