Question

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=

Options

• 9

• 5

• 17

• 7

Answer 1

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`