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प्रश्न
If \[x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\] and \[y = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}\] then x + y +xy=
विकल्प
9
5
17
7
उत्तर
Given that `x=(sqrt5 +sqrt3)/(sqrt5 - sqrt3)`and `y = (sqrt5 - sqrt3)/(sqrt5 +sqrt3)`.
We are asked to find `x+y + xy`
Now we will rationalize x. We know that rationalization factor for `sqrt5 -sqrt3` is `sqrt5 +sqrt3`. We will multiply numerator and denominator of the given expression `x=(sqrt5 +sqrt3)/(sqrt5 - sqrt3)` by `sqrt5 + sqrt3`, to get
`x=(sqrt5 +sqrt3)/(sqrt5 - sqrt3) xx (sqrt5 +sqrt3)/(sqrt5 + sqrt3)`
`= ((sqrt5)^2+(sqrt3)^2+ 2 xx sqrt5 xx sqrt3)/((sqrt5)^2 - (sqrt3)^2)`
`= (5+3+2sqrt15)/(5-3)`
`= 4 + sqrt15`
Similarly, we can rationalize y. We know that rationalization factor for `sqrt5 +sqrt3`is`sqrt5 - sqrt3`. We will multiply numerator and denominator of the given expression `(sqrt5 - sqrt3)/(sqrt5+sqrt3)`by,`sqrt5 - sqrt3` to get
x = `(sqrt5 - sqrt3)/(sqrt5+sqrt3) xx (sqrt5 - sqrt3)/(sqrt5-sqrt3)`
` = ((sqrt5)^2 + (sqrt3)^2 - 2 xx sqrt5 xx sqrt3)/((sqrt5)^2 - (sqrt3)) `
`= (5+3-2sqrt15)/(5-3)`
`= (8-2sqrt15)/2`
`= 4-sqrt15`
Therefore,
`x+y+xy = 4 +sqrt15 + 4 -sqrt15 +(4+sqrt15) (4-sqrt15)`
`= 4+4 + 16 - 4sqrt15 + 4 sqrt15 - (sqrt15)^2`
` = 24 - 15 `
=` 9`
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