Question

1/x+1/y varies as 1/x+y then prove x varies as y​

Answer 1

Variation problem

Given. 1/x + 1/y varies as 1/(x + y)

To find. x varies as y

Solution.

Here, 1/x + 1/y ∝ 1/(x + y)

or, 1/x + 1/y = k * 1/(x + y), where k is constant of variation

or, (x + y) / (xy) = k * 1/(x + y)

or, (x + y)² / (xy) = k

or, (x² + 2xy + y²) / (xy) = k

or, x/y + 2 + y/x = k

or, (x/y)² + 2 (x/y) + 1 = k (x/y) [ multiplying both sides by x/y ]

or, (x/y)² + (2 - k) (x/y) + 1 = 0

or, (x/y)² + c (x/y) + 1 = 0, where c = 2 - k = constant

Then, x/y = {- c ± √(c² - 4)} / 2 = constant = p (say)

or, x/y = p

or, x = py

Since p is constant, we can write

x ∝ y

This completes the proof.