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Mathematics

If a, c, a2 + b2 and b2 + c2 are in proportion, show that b is mean proportion between a and c.

Ratio Proportion

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Answer

Given,

a, c, a2 + b2 and b2 + c2 are in proportion.

∴ca=b2+c2a2+b2⇒c(a2+b2)=a(b2+c2)⇒ca2+cb2=ab2+ac2⇒cb2−ab2=ac2−ca2⇒b2(c−a)=ac(c−a)⇒b2=ac⇒ab=bc.\therefore \dfrac{c}{a} = \dfrac{b^2 + c^2}{a^2 + b^2} \\[1em] \Rightarrow c(a^2 + b^2) = a(b^2 + c^2) \\[1em] \Rightarrow ca^2 + cb^2 = ab^2 + ac^2 \\[1em] \Rightarrow cb^2 - ab^2 = ac^2 - ca^2 \\[1em] \Rightarrow b^2(c - a) = ac(c - a) \\[1em] \Rightarrow b^2 = ac \\[1em] \Rightarrow \dfrac{a}{b} = \dfrac{b}{c}.

Hence, proved that b is the mean proportion between a and c.

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