Find the condition that x3 – px2 + qx – r = 0 may have the roots i

1.	Find the condition that x3 – px2 + qx – r = 0 may have the roots in G.P.
Answer» Let , a/R, a, aR be the roots of x³ – px² + qx – r = 0 Then product of the roots = a/R .a. aR = a³ = r ⇒ a = r^1/3 ∵ a is a root of x³ – px² + qx – r = 0 ⇒ a³ – pa² + qa – r = 0 But a = r^1/3 ⇒ (r^1/3)³ – p(r^1/3)² + q(r^1/3) – r = 0 ⇒ r – p. r^2/3 + q. r^1/3 – r = 0 ⇒ p. r^2/3 = q r^1/3 By cubing on both sides ⇒ p³r² = q³r ⇒ p³r = q³ is the required condition.

Find the condition that x3 – px2 + qx – r = 0 may have the roots in G.P.

Answer»

Let , a/R, a, aR be the roots of x³ – px² + qx – r = 0

Then product of the roots = a/R .a. aR = a³ = r ⇒ a = r^1/3

∵ a is a root of x³ – px² + qx – r = 0

⇒ a³ – pa² + qa – r = 0

But a = r^1/3

⇒ (r^1/3)³ – p(r^1/3)² + q(r^1/3) – r = 0

⇒ r – p. r^2/3 + q. r^1/3 – r = 0 ⇒ p. r^2/3 = q r^1/3

By cubing on both sides

⇒ p³r² = q³r

⇒ p³r = q³ is the required condition.

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Find the condition that x3 – px2 + qx – r = 0 may have the roots in G.P.

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