Question

If $x_1,x_2,x_3$ as well as $y_1,y_2,y_3$ are in geometric progression with the same common ratio, then the points $\left(x_1,y_1\right),\left(x_2,y_2\right),\left(x_3,y_3\right)$ are

• A. vertices of an equilateral triangle
• B. vertices of a right angled triangle
• C. vertices of a right angled isosceles triangle
• D. collinear

Answer 1

Answer: (D)

We have $x_1,x_2,x_3$ and $y_1,y_2,y_3$ are in $GP$ with the same common ratio.
Let $r$ be the common ratio.
$\therefore x_1=x,x_2=xr\text{ and }{x}_3=x{r}^2$
Similarly, $y_1=y$
$y_2=yr$ and $y_3=y{r}^2$
$\therefore$ Area of $\Delta =\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$
$=\frac{1}{2}\left|\begin{array}{ccc}x& y& 1\\ xr& yr& 1\\ x{r}^2& y{r}^2& 1\end{array}\right|$
$=\frac{1}{2}xy\left|\begin{array}{ccc}1& 1& 1\\ r& r& 1\\ r^2& r^2& 1\end{array}\right|=\frac{1}{2}\times 0=0$
$[\because C_1,C_2$ are identical $]$
$\therefore$ The given points are collinear.