Question

If $x_1,x_2,x_{3}and{y}_1,y_2,y_3$ are both in GP with the same common ratio, then the points $(x_1,y_1),(x_2,y_2)and(x_3,y_3)$

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

Answer 1

Answer: (A)

lie on a straight line

Let the common ratio $=r$

$x_1=x_1{x}_2=x_{1}r{x}_3=x_1{r}^2{y}_1=y_1{y}_2=y_{1}r{y}_3=y_1{r}^2$

Slope of $(x_1,y_1)$ and $(x_2,y_2)=\frac{y_2-y_1}{x_2-x_1}=\frac{y_1(r-1)}{x_1(r-1)}=\frac{y_1}{x_1}$

Slope of $(x_2,y_2),(x_3,y_3)=\frac{y_3-y_2}{x_3-x_2}=\frac{y_1(r-1)}{x_1(r-1)}=\frac{y_1}{x_1}$

Both have same slope

Hence they lie on straight line.