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

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

Answer 1

Answer: (A)

Lie on a straight line

$x_1,x_2,x_3⟶G.P$

Let common ratio be r

$\therefore x_2=x_{1}r{x}_3=x_2{r}^2{y}_{1,}{y}_2{,y}_3\to G.P{y}_2=y_{1}r{y}_3=y_1{r}^{2}P\left(x_1{,y}_1\right)Q\left(x_2{,y}_2\right)R\left(x_3{,y}_3\right)$

$SlopeofPQ=\frac{{y_2-y}_1}{x_2-x_1}=\frac{y_{1}r-y_1}{x_{1}r-x_1}=\frac{y_1}{x_1}$

$SlopeofQR=\frac{{y_3-y}_2}{x_3-x_2}=\frac{y_1{r}^2-y_{1}r}{{x_1{r}^2-x}_{1}r}=\frac{y_1}{x_1}$

As slopes are equal so P,Q,R are collinear and they lie on a straight line.