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Question
Find the mean proportional of the following:
`(a - b)/(a + b) and (a^2b^2)/(a^2 - b^2)`
Sum
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Solution
Let x be the mean proportional between `(a - b)/(a + b) and (a^2b^2)/(a^2 - b^2)`
Then,
`(a - b)/(a + b)` : x :: x : `(a^2b^2)/(a^2 - b^2)`
⇒ `(a - b)/(a + b)` : x = x : `(a^2b^2)/(a^2 - b^2)`
⇒ `[(a - b)/(a + b)]/x = x/[(a^2b^2)/(a^2 - b^2)]`
⇒ x × x = `(a - b)/(a + b) xx (a^2b^2)/(a^2 - b^2)`
⇒ x2 = `(a - b)/(a + b) xx (a^2b^2)/[(a - b)(a + b)]`
⇒ x2 = `1/(a + b) xx (a^2b^2)/(a + b)`
⇒ x2 = `(a^2b^2)/(a + b)^2`
⇒ x = `sqrt[(a^2b^2)/(a + b)^2]`
⇒ x = `(ab)/(a + b)`
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Chapter 7: Ratio and proportion - Exercise 7B [Page 125]
