Question

If $x_1,x_2,x_3$ as well as $y_1,y_2,y_3$ are in G.P. with same common ratio, then the points $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$

• A. are collinear
• B. lie on a circle
• C. lie on a triangle
• D. nothing can be said

Answer 1

The correct option is A are collinear

Since $x_1,x_2,x_3$ and $y_1,y_2,y_3$ are in G.P. with same common ratio, we can consider them as: $x_2=x_{1}r,x_3=x_1{r}^2$ and similarly, $y_2=y_{1}r,y_3=y_1{r}^2$
$\Delta =\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ r{x}_1& r{y}_1& 1\\ r^2{x}_1& r^2{y}_1& 1\end{array}\right|=\frac{1}{2}{x}_1{y}_1\left|\begin{array}{ccc}1& 1& 1\\ r& r& 1\\ r^2& r^2& 1\end{array}\right|=0$
Hence the points are collinear