Question

If a₁, a₂, a₃ and b₁, b₂, b₃ are in geometric progression and their common ratios are equal, then the points A(a₁, b₁), B(a₂, b₂) and C(a₃, b₃) …..

(a) lie on the same line

(b) lie on a circle

(c) lie on an ellipse

(d) None of these

Answer 1

The correct option (a) lie on the same line

Explanation:

Let r be the common ratio

∴ a₂ = a₁r,

a₃ = a₁r²,

b₂ = b₁r

b₃ = b₁r²

slope of vector AB = [(b₂ – b₁)/(a₂ – a₁)] = [{b₁(1 – r)}/{a₁(1 – r)}] = (b₁/a₁)

slope of vector BC = [(b₃ – b₂)/(a₃ – a₂)] = (b₁/a₁)

∵ slope of vector AB = slope of BC

∴ A,B, C are on a line.