QQuestion
If (√a + √b) ∝ (√a-√b) then show that (a + b) ∝ √(ab)
✓Answer
(√a + √b) ∝ (√a-√b)
⇒ (√a + √b) = k (√a-√b) (where k = non-zero constant)
⇒ \frac{√a + √b}{√a - √b} = k
Squaring both sides
⇒ \frac{(√a + √b)²}{(√a - √b)²} = k²
Applying Componendo and Dividendo
⇒ \frac{(√a + √b)² + (√a - √b)²}{(√a + √b)² - (√a - √b)²} = k² + 1\over k² - 1
⇒ \frac{a + 2√ab + b + a - 2√ab + b}{a + 2√ab + b - a + 2√ab - b} = k² + 1\over k² - 1
⇒ \frac{2(a + b)}{4√ab} = k² + 1\over k² - 1
⇒ \frac{(a + b)}{2√ab} = k² + 1\over k² - 1
⇒ (a + b) = k² + 1\over k² - 1 × 2√ab
⇒ (a + b) ∝ √ab (Hence Proved)
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