Question

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

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

Answer 1

The correct option is A

Lie on a straight line

Explanation for the correct option :

As the common ratio of $x_1,x_2,x_3$ is same as $y_1,y_2,y_3$, so they can be written as

$x_1=a,x_2=ar,x_3=a{r}^{2}and{y}_1=b,y_2=br,y_3=b{r}^2$

So, the points will be

$$\begin{gathered} P(x_1,y_1)=P\left(a,b\right), \\ Q(x_2,y_2)=Q\left(ar,br\right)\mathrm{and} \\ R(x_3,y_3)=R\left(a{r}^2,b{r}^2\right) \end{gathered}$$

Now, let us find the slope,

Slope of $PQ=\frac{br-b}{ar-a}=\frac{b}{a}$ [by slope of line $=\frac{y_2-y_1}{x_2-x_1}$]

Slope of $QR=\frac{b{r}^2-br}{a{r}^2-ar}=\frac{b}{a}$

As the slopes are same and $Q$ is a common point, that means the points are collinear and thus they lie on a straight line.

Hence, option A is correct.