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Question
If a, b, c, d are in proportion, then prove that
`sqrt[( a^2 + 5c^2)/( b^2 + 5d^2)] = a/b`
Solution
It is given that a, b, c, d are in proportion.
`therefore a/b = c/d = k`
⇒ a = bk, c = dk
`sqrt[( a^2 + 5c^2)/( b^2 + 5d^2)]`
= `sqrt[(bk)^2 + 5(dk)^2]/(b^2 + 5d^2 )`
= `sqrt[k^2( b^2 + 5d^2)]/(b^2 + 5d^2)`
= `sqrt(k^2)`
= k ...(1)
`a/b = [bk]/b = k` ...(2)
From (1) and (2), we get
`sqrt([a^2 + 5c^2]/[b^2 + 5d^2]) = a/b`
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