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Question
Find the value of a and b in the following:
`(7 + sqrt(5))/(7 - sqrt(5)) - (7 - sqrt(5))/(7 + sqrt(5)) = a + 7/11 sqrt(5)b`
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Solution
We have, `(7 + sqrt(5))/(7 - sqrt(5)) - (7 - sqrt(5))/(7 + sqrt(5)) = a + 7/11 sqrt(5)b`
⇒ `((7 + sqrt(5))^2 - (7 - sqrt(5))^2)/((7 - sqrt(5))(7 + sqrt(5))) = a + 7/11 sqrt(5)b`
⇒ `([7^2 + (sqrt(5))^2 + 2 xx 7 xx sqrt(5)] - [7^2 + (sqrt(5))^2 - 2 xx 7 xx sqrt(5)])/(7^2 - (sqrt(5))^2) = a + 7/11 sqrt(5)b`
⇒ `(49 + 5 + 14sqrt(5) - 49 - 5 + 14sqrt(5))/(49 - 5) = a + 7/11 sqrt(5)b` ...`[("Using identity" (a + b)^2 = a^2 + 2ab + b^2),((a - b)^2 = a^2 - 2ab - b^2),("and" (a - b)(a + b) = a^2 - b^2)]`
⇒ `(28sqrt(5))/44 = a + 7/11 sqrt(5)b`
⇒ `7/11 sqrt(5) = a + 7/11 sqrt(5)b`
⇒ `0 + 7/11 sqrt(5) = a + 7/11 sqrt(5)b`
On comparing both sides, we get
a = 0 and b = 1
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