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 prove that the points `(x_(1),y_(1)),(x_(2),y_(2))` and `(x_(3),y_(3))` are collinear.

Options

A. equal in area
B. similar
C. congruent
D. none of these

Answer

A

Explanation

We have
`(1)/(2)|{:(x_1,,y_1,,1),(x_2,,y_2,,1),(x_3,,y_3,,1):}|=|{:(a_1,,b_1,,1),(a_2,,b_2,,1),(a_3,,b_3,,1):}|or (1)/(2)|{:(x_1,,y_1,,1),(x_2,,y_2,,1),(x_3,,y_3,,1):}|=(1)/(2)|{:(a_1,,b_1,,1),(a_2,,b_2,,1),(a_3,,b_3,,1):}|`
Hence, the area of triangle with vertices `(x_1,y_1),(x_2,y_2),(x_3,x_3)` is the same as the area of triangle with vertices `(a_1,b_1),(a_2,b_2),(a_3,b_3)`. Hence, the two triangles are equal in area.