Advertisements
Advertisements
Question
If `a = (4sqrt6)/(sqrt2 + sqrt3)`, find the value of `(a + 2sqrt2)/(a - 2sqrt2) + (a + 2sqrt3)/(a - 2sqrt3)`.
Solution
`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`
APPEARS IN
RELATED QUESTIONS
If `x = (6ab)/(a + b)`, find the value of `(x + 3a)/(x - 3a) + (x + 3b)/(x - 3b)`.
If a : b : : c : d, then prove that
`(4"a" + 9"b")/(4"c" + 9"d") = (4"a" - 9"b")/(4"c" - 9"d")`
If p, q, r ands are In continued proportion, then prove that (p3+q3+r3) ( q3+r3+s3) : : P : s
Find x, if `16((a - x)/(a + x))^3 = (a + x)/(a - x)`.
Show, that a, b, c, d are in proportion if:
(6a + 7b) : (6c + 7d) : : (6a - 7b) : (6c - 7d)
Given : x = `(sqrt(a^2 + b^2)+sqrt(a^2 - b^2))/(sqrt(a^2 + b^2)-sqrt(a^2 - b^2))`
Use componendo and dividendo to prove that `b^2 = (2a^2x)/(x^2 + 1)`.
If `(8a - 5b)/(8c - 5a) = (8a + 5b)/(8c + 5d)`, prove that `a/b = c/d`
Solve `(1 + x + x^2)/(1 - x + x^2) = (62(1 +x))/(63(1 + x)`
If x = `(root(3)(a + 1) + root(3)(a - 1))/(root(3)(a + 1) - root(3)(a - 1)`,prove that :
x³ – 3ax² + 3x – a = 0
If `(x^2 - 4)/(x^2 + 4) = 3/5`, the value of x is ______.
