Suppose that the three points $A, B$ and $C$ in the plane are such that their $x$-coordinates as well as…
Suppose that the three points $A, B$ and $C$ in the plane are such that their $x$-coordinates as well as $y$-coordinates are in GP with the same common ratio. Then, the points $A, B$ and $C$
constitute a right angled triangle
form an isosceles triangle
lie on a straight line
form an equilateral triangle
Solution
Let the coordinate $A\left(x_1, y_1\right), B\left(x_2, y_2\right)$ and $C\left(x_3, y_3\right)$.
According to question, $x$ - coordinate as well as $y$ coordinates are in GP.
Let $x_1=a, x_2=a r$ and $x_3=a r^2$,
$y_1=b, y_2=b r$ and $y_3=b r^2$
Now, $A(a, b), B(a r, b r), C\left(a r^2, b r^2\right)$.
Slope of $A B=\frac{b(\mathrm{l}-r)}{a(\mathrm{l}-r)}=\frac{b}{a}$ and
Slope of $\mathrm{BC}=\frac{b r(1-r)}{\operatorname{ar}(1-r)}=\frac{b}{a}$
As slope of $A B=$ Slope of $B C$
$\therefore A, B$ and $C$ are collinear.
$\therefore A, B$ and $C$ lie on a straight line.