Advertisements
Advertisements
Question
If `x = (sqrt(a + 3b) + sqrt(a - 3b))/(sqrt(a + 3b) - sqrt(a - 3b))`, prove that: 3bx2 – 2ax + 3b = 0.
Advertisements
Solution
`x = (sqrt(a + 3b) + sqrt(a - 3b))/(sqrt(a + 3b) - sqrt(a - 3b))`
Applying componendo and dividendo, we get,
`(x + 1)/(x - 1) = (sqrt(a + 3b) + sqrt(a - 3b) + sqrt(a + 3b) - sqrt(a - 3b))/(sqrt(a + 3b) + sqrt(a - 3b) - sqrt(a + 3b) + sqrt(a - 3b))`
`(x + 1)/(x - 1)= (2sqrt(a + 3b))/(2sqrt(a - 3b))`
Squaring both sides,
`(x^2 + 2x + 1)/(x^2 - 2x + 1) = (a + 3b)/(a - 3b)`
Again applying componendo and dividendo,
`(x^2 + 2x + 1 + x^2 - 2x + 1)/(x^2 + 2x + 1 - x^2 + 2x - 1) = (a + 3b + a - 3b)/(a + 3b - a + 3b)`
`(2(x^2 + 1))/(2(2x)) = (2(a))/(2(3b))`
3b(x2 + 1) = 2ax
3bx2 + 3b = 2ax
3bx2 – 2ax + 3b = 0
RELATED QUESTIONS
If y is the mean proportional between x and y; show that y(x+z) is the mean p roporti ona I between x2+ y2 and y2+ z2
If a, b, c and dare in continued proportion, then prove that
(a+ d)(b+ c)-(a+ c)(b+ d)= (b-c)2
Find the value of x in the following proportions : 2.5 : 1.5 = x : 3
Find the value of x in each of the following proportions:
x : 92 : : 87 : 116
Determine if the following ratio form a proportion:
39 litres : 65 litres and 6 bottles : 10 bottles
4.5 g of an alloy of copper and zinc contains 3.5 g of copper. What weight of copper will there be in 18.9 g of the alloy?
The cost of 4 dozen bananas is Rs 104. How many bananas can be purchased for Rs 6.50?
A given quantity of rice is sufficient for 60 persons for 3 days. How many days would the rice last for 18 persons?
If `a/c = c/d = c/f` prove that : `bd f[(a + b)/b + (c + d)/d + (c + f)/f]^3` = 27(a + b)(c + d)(e + f)
If a, b, c, d are in continued proportion, prove that: (a + d)(b + c) – (a + c)(b + d) = (b – c)2
