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प्रश्न
If `a = (4sqrt6)/(sqrt2 + sqrt3)`, find the value of `(a + 2sqrt2)/(a - 2sqrt2) + (a + 2sqrt3)/(a - 2sqrt3)`.
उत्तर
`a = (4sqrt6)/(sqrt2 + sqrt3)`
`a/(2sqrt2) = (2sqrt3)/(sqrt2 + sqrt3)`
Applying componendo and dividendo,
`(a + 2sqrt2)/(a - 2sqrt2) = (2sqrt3 + sqrt2 + sqrt3)/(2sqrt3 - sqrt2 - sqrt3)`
`(a + 2sqrt2)/(a - 2sqrt2) = (3sqrt3 + sqrt2)/(sqrt3 - sqrt2)` ...(1)
`a/(2sqrt3) = (2sqrt2)/(sqrt2 + sqrt3)`
Applying componendo and dividendo,
`(a + 2sqrt3)/(a - 2sqrt3) = (2sqrt2 + sqrt2 + sqrt3)/(2sqrt2 - sqrt2 - sqrt3)`
`(a + 2sqrt3)/(a - 2sqrt3) = (3sqrt2 + sqrt3)/(sqrt2 - sqrt3)` ...(2)
From (1) and (2),
`(a + 2sqrt2)/(a - 2sqrt2) + (a + 2sqrt3)/(a - 2sqrt3) = (3sqrt3 + sqrt2)/(sqrt3 - sqrt2) + (3sqrt2 + sqrt3)/(sqrt2 - sqrt3)`
`(a + 2sqrt2)/(a - 2sqrt2) + (a + 2sqrt3)/(a - 2sqrt3) = (3sqrt2 + sqrt3 - 3sqrt3 - sqrt2)/(sqrt2 - sqrt3)`
`(a + 2sqrt2)/(a - 2sqrt2) +(a + 2sqrt3)/(a - 2sqrt3) = (2sqrt2 - 2sqrt3)/(sqrt2 - sqrt3) = 2`
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