IPMAT Indore 2023 - QA (MCQ) Q34 Explanation

Explanation:

(a)

∵ a1, a2, a3 are in geometric progression.

a2/a1 = a3/a2 = r

a2 = a1r − (i)

a3 = a2r = a1r2 −(ii)

Given: a1 x^2 + 2a2x + a3 = 0

Putting the values of a2 and a3 in above equation, we get

a1x^2 + 2a1rx + a1r^2 = 0

a1(x^2 + 2r⋅x + r^2) = 0

a1(x+r)^2 = 0

a1 ≠ 0 ∴ x+r = 0

x = −r.

Let -r be the common root of given two equations.

Then b1x^2 + 2b2x + b3 = 0 can be written as:

b1r^2 − 2b2r + b3 = 0

b1r^2 + b3 = 2b2r from −(ii)

b1. a3/a1 + b3= 2b2⋅ a3/a2

b1/a3 a3/a1 + b3/a3 = 2b2/a3 a3/a2

b1/a1 + b3/a3 = 2b2/a2

b1/a1 , b2/a2 , b3/a3 are in arithmetic progression. Ans.

Note : If x, y, z are in A.P. then x+z=2y