Question

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

• A. are vertices of a triangle
• B. lie on a straight line
• C. lie on an ellipse
• D. lie on a circle

Answer 1

Answer: (B)

Taking co-ordinates as
$\left(\frac{x}{r} , \frac{y}{r}\right) ; \left(x,y\right) \& \left(xr , yr\right). $
Then slope of line joining
$ \left(\frac{x}{r} , \frac{y}{r} \right) , \left(x,y\right) = \frac{y\left(1- \frac{1}{r}\right)}{x\left(1- \frac{1}{r}\right)} = \frac{y}{x}$
and slope of line joining (x, y) and (xr, yr)
$ = \frac{y\left(r-1\right)}{x\left(r-1\right)} = \frac{y}{x}$
$ \therefore m_{1} =m_{2}$
$\Rightarrow $ Points lie on the straight line.