Question

If $x_1,x_{2}and{x}_3$ as well as $y_1,y_{2}and{y}_3$ are in GP with the same common ratio, the points
$P(x_1,y_1),Q(x_2,y_2)andR(x_3,y_3)$

• A. lie on a straight line
• B. lie on a 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 the common ratio be r and assume
$x_1=a,x_2=ar,x_3=a{r}^2$
And $y_1=b,y_2=br,y_3=b{r}_2$
If we observe the given points, we find that
Slope of PQ = Slope of QR.
$Slope=\frac{y_2-y_1}{x_2-x_1}$
$\frac{br-b}{ar-a}=\frac{b{r}^2-br}{a{r}^2-ar}$
$\frac{b}{a}=\frac{b}{a}$
Slope of PQ=Slope of QR
Hence, we can say P,Q,R are collinear points or pqr lie on a straight line.