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Mathematics

Find the mean proportional between a - b and a3 - a2b

Ratio Proportion

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Answer

Let mean proportional between a - b and a3 - a2b be x

⇒a−bx=xa3−a2b⇒x2=(a−b)(a3−a2b)⇒x2=a4−a3b−a3b+a2b2⇒x2=a4−2a3b+a2b2⇒x2=a2(a2+b2−2ab)⇒x2=a2(a−b)2⇒x=a(a−b).\Rightarrow \dfrac{a - b}{x} = \dfrac{x}{a^3 - a^2b} \\[1em] \Rightarrow x^2 = (a - b)(a^3 - a^2b) \\[1em] \Rightarrow x^2 = a^4 - a^3b - a^3b + a^2b^2 \\[1em] \Rightarrow x^2 = a^4 - 2a^3b + a^2b^2 \\[1em] \Rightarrow x^2 = a^2(a^2 + b^2 - 2ab) \\[1em] \Rightarrow x^2 = a^2(a - b)^2 \\[1em] \Rightarrow x = a(a - b).

Hence, mean proportional between a - b and a3 - a2b = a(a - b).

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