Question

Three points $ A, B, C $ in the plane have: - x-coordinates in G.P. - y-coordinates in the same G.P. Then, the points $ A, B, C $ lie on:

• A. A right-angled triangle
• B. An isosceles triangle
• C. A straight line
• D. An equilateral triangle

Hint:

If x and y coordinates of points are in geometric progression with the same ratio, the points lie on a straight line.

Answer 1

The Correct Option is C

Let the x-coordinates be: \[ a, ar, ar^2 \] Let the y-coordinates also be in same G.P.: \[ b, br, br^2 \] So points are:
- \( A = (a, b) \)
- \( B = (ar, br) \)
- \( C = (ar^2, br^2) \) Observe: \[ \text{Slope of AB} = \frac{br - b}{ar - a} = \frac{b(r - 1)}{a(r - 1)} = \frac{b}{a} \] \[ \text{Slope of BC} = \frac{br^2 - br}{ar^2 - ar} = \frac{br(r - 1)}{ar(r - 1)} = \frac{b}{a} \] Since slope AB = slope BC ⇒ points lie on the same line. Final Conclusion: \[ \boxed{A, B, C \text{ are collinear}} \]