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Mathematics

If b is the mean proportion between a and c, prove that :

a4+a2b2+b4b4+b2c2+c4=a2c2.\dfrac{a^4 + a^2b^2 + b^4}{b^4 + b^2c^2 + c^4} = \dfrac{a^2}{c^2}.

Ratio Proportion

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Answer

Given,

b is the mean proportion between a and c.

∴ab=bc\therefore \dfrac{a}{b} = \dfrac{b}{c}

⇒ b2 = ac ………(1)

Solving, L.H.S. of the equation a4+a2b2+b4b4+b2c2+c4=a2c2\dfrac{a^4 + a^2b^2 + b^4}{b^4 + b^2c^2 + c^4} = \dfrac{a^2}{c^2}, we get :

⇒a4+a2b2+b4b4+b2c2+c4⇒a4+a2b2+(b2)2(b2)2+b2c2+c4⇒a4+a2b2+(ac)2(ac)2+b2c2+c4 [From equation (1)]⇒a4+a2b2+a2c2a2c2+b2c2+c4⇒a2(a2+b2+c2)c2(a2+b2+c2)⇒a2c2.\Rightarrow \dfrac{a^4 + a^2b^2 + b^4}{b^4 + b^2c^2 + c^4} \\[1em] \Rightarrow \dfrac{a^4 + a^2b^2 + (b^2)^2}{(b^2)^2 + b^2c^2 + c^4} \\[1em] \Rightarrow \dfrac{a^4 + a^2b^2 + (ac)^2}{(ac)^2 + b^2c^2 + c^4} \text{ [From equation (1)]}\\[1em] \Rightarrow \dfrac{a^4 + a^2b^2 + a^2c^2}{a^2c^2 + b^2c^2 + c^4}\\[1em] \Rightarrow \dfrac{a^2(a^2 + b^2 + c^2)}{c^2(a^2 + b^2 + c^2)} \\[1em] \Rightarrow \dfrac{a^2}{c^2}.

Since, L.H.S. = R.H.S.

Hence, proved that a4+a2b2+b4b4+b2c2+c4=a2c2\dfrac{a^4 + a^2b^2 + b^4}{b^4 + b^2c^2 + c^4} = \dfrac{a^2}{c^2}.

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