If a + b varies directly with a - b, then which of the following vary directly?

Options

A. a and b
B. `a^(2) + b^(2) and a^(2)-b^(2)`
C. both (a) and (b)
D. None of these

Answer

C

Explanation

Given that a + b vary directly with a - b.
` (a + b)/(a - b) = k` (constant)
Applying componendo - dividendo theorem,
`a/b = (k+1)/(k-1)` as k is constant, `(k+1)/(k-1)` is also a constant.
` :. ` a and b vary directly.
Let ` (k+1)/(k-1) ` be p
` rArr a/b = p rArr a^(2)/b^(2) = p^(2)`.
By applying componendo-dividendo theorem,
` (a^(2)+b^(2))/(a^(2)-b^(2)) = (p^(2)+1)/(p^(2)-1)`.
As p is a constant, `(p^(2)+1)/(p^(2)-1)` is also a consonant.
` :. a^(2) + b^(2) and a^(2) - b^(2)` vary directly.