Advertisements
Advertisements
Question
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)`
Solution
Given: `x/(1) = (sqrt(a^2 + b^2) + sqrt(a^2 - b^2))/(sqrt(a^2 + b^2) - sqrt(a^2 - b^2)`
Applying componendo and dividendo both sides, we have
`(x + 1)/(x - 1) = (sqrt(a^2 + b^2) + sqrt(a^2 - b^2) + sqrt(a^2 + b^2) - sqrt(a^2 - b^2))/(sqrt(a^2 + b^2) + sqrt(a^2 - b^2) - sqrt(a^2 + b^2) + sqrt(a^2 - b^2)`
`\implies (x + 1)/(x - 1) = (2sqrt(a^2 + b^2))/(2sqrt(a^2 + b^2)`
`\implies (x + 1)/(x - 1) = (sqrt(a^2 + b^2))/(sqrt(a^2 + b^2)`
Squaring both sides, we have
`\implies (x + 1)^2/(x - 1)^2 = (a^2 + b^2)/(a^2 - b^2)`
`\implies (x^2 + 1 + 2x)/(x^2 + 1 - 2x) = (a^2 + b^2)/(a^2 - b^2)`
Applying componendo and dividendo both sides, we get
`\implies (x^2 + 1 + 2x + x^2 + 1 - 2x)/(x^2 + 1 + 2x - x^2 - 1 + 2x) = (a^2 + b^2 + a^2 - b^2)/(a^2 + b^2 - a^2 + b^2)`
`\implies (2x^2 + 2)/(4x) = (2a^2)/(2b^2)`
`\implies (x^2 + 1)/(2x) = a^2/b^2`
`\implies b^2 = (2a^2x)/(x^2 + 1)`
APPEARS IN
RELATED QUESTIONS
If a : b = c : d, prove that: 5a + 7b : 5a – 7b = 5c + 7d : 5c – 7d.
If (7a + 8b)(7c – 8d) = (7a – 8b)(7c + 8d); prove that a : b = c : d.
If p, q, r ands are In continued proportion, then prove that (p3+q3+r3) ( q3+r3+s3) : : P : s
If x = `(root (3)("m + 1") + root (3)("m - 1"))/(root (3)("m + 1") + root (3)("m - 1")` then prove that x3 - 3mx2 + 3x = m
If y = `((p + 1)^(1/3) + (p - 1)^(1/3))/((p + 1)^(1/3) - (p - 1)^(1/3)` find that y3 - 3py2 + 3y - p = 0.
Using componendo and idendo, find the value of x
`(sqrt(3x + 4) + sqrt(3x - 5))/(sqrt(3x + 4) - sqrt(3x - 5)` = 9
if `(3a + 4b)/(3c + 4d) = (3a - 4b)/(3c - 4d)` Prove that `a/b = c/d`.
If x = `(4sqrt(6))/(sqrt(2) + sqrt(3)` find the value of `(x + 2sqrt(2))/(x - 2sqrt(2)) + (x + 2sqrt(3))/(x - 2sqrt(3)`
If x = `(2mab)/(a + b)`, find the value of `(x + ma)/(x - ma) + (x + mb)/(x - mb)`
If `(x^2 - 1)/(x^2 + 1) = 3/5`, the value of x is ______.
