If `x_1, x_2, x_3` as well as `y_1, y_2, y_3` are 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) lie on a straight line (B) lie on a parabola (C) lie on a circle (D) are vertices of a triangle

Options

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

Answer

A

Explanation

Let r be the common ratio. Then ,
`x_(2)=x_(1)r,x_(3)=x_(1)r^(2), y_(2)=y_(1)r and y_(3)=y_(1)r^(2)`
`therefore (y_(2)-y_(1))/(x_(2)-x_(1))=(y_(1)r-y_(1))/(x_(1)r-x_(1))=(y_(1))/(x_(1))`
and , `(y_(3)-y_(2))/(x_(3)-x_(2))=(y_(1)r^(2)-y_(1)r)/(x_(1)r^(2)-x_(1)r)=(y_(1))/(x_(1))`
So, slope of PQ, Slope of QR
Hence , points P, Q, R are collinear.