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 point $P(x_1,y_1),Q(x_2,y_2)$ and $R(x_3,y_3)$

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

Answer 1

The correct option is A lie on a straight line

Let $P(x_1,y_1)Q(x_2,y_2)R(x_3,y_3)$

Let a be the first term and be the common ratio of first G.P. then

$x_1=a,x_2=ar{x}_3={ar}^2$

Let b be the first term and r be the common term,ratio of second G.P,then

$\frac{y_2-y_1}{x_2-x_1}=\frac{br-b}{ar-a}=\frac{b}{a}=$ Slope of PQ

$\frac{y_3-y_2}{x_3-x_2}=\frac{b{r}^2-b}{a{r}^2-r}=\frac{b}{a}=$ Slope of QR

$\therefore$ Slope of $PQ$ = slope of $QR$

$\therefore$ P,Q and R lie on a straight line