Question

Suppose that the three points \(A\), \(B\) and \(C\) in the plane are such that their \(x\)-coordinates as well as \(y\)-coordinates are in G.P. with the same common ratio. Then the points \(A\), \(B\) and \(C\)

• A. constitute a right angled triangle
• B. form an isosceles triangle
• C. lie on a straight line
• D. form an equilateral triangle

Hint:

If the slopes between consecutive pairs of points are equal, then all the points are collinear and lie on the same straight line.

Answer 1

The Correct Option is C

Step 1: Represent the coordinates using G.P.
Let the coordinates of the points be: \[ A(x_1,y_1),\quad B(x_2,y_2),\quad C(x_3,y_3) \] Since the \(x\)-coordinates are in G.P. with common ratio \(r\), \[ x_2=x_1r,\qquad x_3=x_1r^2 \] Similarly, since the \(y\)-coordinates are also in G.P. with the same common ratio \(r\), \[ y_2=y_1r,\qquad y_3=y_1r^2 \] Thus, \[ A=(x_1,y_1) \] \[ B=(x_1r,y_1r) \] \[ C=(x_1r^2,y_1r^2) \]

Step 2: Find the slope of \(AB\).
\[ m_{AB} = \frac{y_1r-y_1}{x_1r-x_1} \] \[ = \frac{y_1(r-1)}{x_1(r-1)} \] \[ = \frac{y_1}{x_1} \]

Step 3: Find the slope of \(BC\).
\[ m_{BC} = \frac{y_1r^2-y_1r}{x_1r^2-x_1r} \] \[ = \frac{y_1r(r-1)}{x_1r(r-1)} \] \[ = \frac{y_1}{x_1} \]

Step 4: Compare the slopes.
Since \[ m_{AB}=m_{BC}, \] the points \(A\), \(B\), and \(C\) are collinear.
Hence, they lie on the same straight line.

Step 5: Final conclusion.
Therefore, \[ \boxed{\text{The points }A,B,C\text{ lie on a straight line}} \]