Given that a3+3ab2b3+3a2b=6362.\dfrac{a^3 + 3ab^2}{b^3 + 3a^2b} = \dfrac{63}{62}.b3+3a2ba3+3ab2=6263. Using componendo and dividendo, find a : b.
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Given,
a3+3ab2b3+3a2b=6362\dfrac{a^3 + 3ab^2}{b^3 + 3a^2b} = \dfrac{63}{62}b3+3a2ba3+3ab2=6263
By componendo and dividendo,
⇒a3+3ab2+b3+3a2ba3+3ab2−b3−3a2b=63+6263−62⇒(a+ba−b)3=125⇒(a+ba−b)3=(5)3⇒a+ba−b=5⇒a+b=5a−5b⇒5a−a=b+5b⇒4a=6b⇒ab=64=23⇒a:b=3:2.\Rightarrow \dfrac{a^3 + 3ab^2 + b^3 + 3a^2b}{a^3 + 3ab^2 - b^3 - 3a^2b} = \dfrac{63 + 62}{63 - 62} \\[1em] \Rightarrow \Big(\dfrac{a + b}{a - b}\Big)^3 = 125 \\[1em] \Rightarrow \Big(\dfrac{a + b}{a - b}\Big)^3 = (5)^3 \\[1em] \Rightarrow \dfrac{a + b}{a - b} = 5 \\[1em] \Rightarrow a + b = 5a - 5b \\[1em] \Rightarrow 5a - a = b + 5b \\[1em] \Rightarrow 4a = 6b \\[1em] \Rightarrow \dfrac{a}{b} = \dfrac{6}{4} = \dfrac{2}{3} \\[1em] \Rightarrow a : b = 3 : 2.⇒a3+3ab2−b3−3a2ba3+3ab2+b3+3a2b=63−6263+62⇒(a−ba+b)3=125⇒(a−ba+b)3=(5)3⇒a−ba+b=5⇒a+b=5a−5b⇒5a−a=b+5b⇒4a=6b⇒ba=46=32⇒a:b=3:2.
Hence, the value of a : b = 3 : 2.
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