Question

If $x_1,x_2,x_3$ and $y_1,y_2,y_3$ are both in G.P. with the same common ratio, then the points $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$

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

Answer 1

Answer: (B)

Taking co-ordinates as
$\left(\frac{x}{r},\frac{y}{r}\right);\left(x,y\right)\&\left(xr,yr\right).$
Then slope of line joining
$\left(\frac{x}{r},\frac{y}{r}\right),\left(x,y\right)=\frac{y\left(1-\frac{1}{r}\right)}{x\left(1-\frac{1}{r}\right)}=\frac{y}{x}$
and slope of line joining (x, y) and (xr, yr)
$=\frac{y\left(r-1\right)}{x\left(r-1\right)}=\frac{y}{x}$
$\therefore m_1=m_2$
$\Rightarrow$ Points lie on the straight line.