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Mathematics

If (4a + 9b)(4c - 9d) = (4a - 9b)(4c + 9d), prove that : a : b = c : d.

Ratio Proportion

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Answer

Given,

⇒(4a+9b)(4c−9d)=(4a−9b)(4c+9d)⇒4a+9b4a−9b=4c+9d4c−9d\Rightarrow (4a + 9b)(4c - 9d) = (4a - 9b)(4c + 9d) \\[1em] \Rightarrow \dfrac{4a + 9b}{4a - 9b} = \dfrac{4c + 9d}{4c - 9d}

Applying componendo and dividendo:

⇒4a+9b+4a−9b4a+9b−(4a−9b)=4c+9d+4c−9d4c+9d−(4c−9d)⇒8a18b=8c18d⇒ab=cd.\Rightarrow \dfrac{4a + 9b + 4a - 9b}{4a + 9b - (4a - 9b)} = \dfrac{4c + 9d + 4c - 9d}{4c + 9d - (4c - 9d)} \\[1em] \Rightarrow \dfrac{8a}{18b} = \dfrac{8c}{18d} \\[1em] \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}.

Hence, proved that a : b = c : d.

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