Question

If $x_1,x_2,x_3$ and $y_1,y_2,y_3$ are in GP with same common ratio, then $(x_1,y_1),(x_2,y_2),(x_3,y_3)$

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

Answer 1

Answer: (D)

lie on a straight line

Let $x_1=a\therefore x_2=ar,x_3=a{r}^2$ and

$y_1=b$ $\therefore y_2=br,y_3=b{r}^2$

Now $A(a,b),B(ar,br),C(a{r}^2,b{r}^2)$

Now slope of $AB=\frac{b(1-r)}{a(1-r)}=\frac{b}{a}$ and

slope of $BC=\frac{br(1-r)}{ar(1-r)}=\frac{b}{a}$

as slope of $AB=$ slope of $BC$

$\therefore AB\parallel BC$ but point $B$ is common so

$A,B,C$ are collinear.