CBSE Class 10 Answered
Let α, β and γ be the roots,
comparing x3 +3px2 +3qx + r with, ax3 +bx2 +cx + d
α + β + γ = -b/a = -3p
αβ+ βγ + αγ = c/a = 3q
αβγ = -d/a = -r,
Now as α, β, γ are required to be in AP.
Let α = a-d, β = a, γ = a+d
α + β + γ = -3p
a-d + a + a + d = -3p
3a = -3p,
a = -p
αβ+ βγ + αγ = 3q
(a-d)a + a(a+d) + (a-d)(a+d) = 3q
a2 - ad + a2 + ad + a2 - d2 = 3q
3a2 - d2 = 3q
3p2 - d2 = 3q
d2 =3p2 - 3q
αβγ = -r
(a-d)(a)(a+d) = -r
a(a2-d2) = -r
-p(p2-d2) = -r
(p2- 3p2 + 3q) = r/p
(3q - 2p2) = r/p
The required condition.
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