Question

Find the ratio of a and b, if a+b/√ab=6

Answer 1

Solution :

Given, (a + b)/√(ab) = 6

  ⇒ a+b = 6 √(ab)

  ⇒ (a+b)² = 36ab .....(i)

Now, (a - b)²

  = (a + b)² - 4ab

  = 36ab - 4ab

  = 32ab

  ⇒ (a - b)² = 32ab .....(ii)

Now, dividing (i) by (ii), we get

$\frac{(a + b)^{2}}{(a - b)^{2}} = \frac{36}{32} = \frac{9}{8}$

$\to \frac{a+b}{a-b} = \frac{3}{2\sqrt{2}}$

$\to \frac{(a+b)+(a-b)}{(a+b)-(a-b)} = \frac{3+2\sqrt{2}}{3-2\sqrt{2}}$

$\to \frac{a+b+a-b}{a+b-a+b} = \frac{3+2\sqrt{2}}{3-2\sqrt{2}}$

$\to \frac{2a}{2b} = \frac{3+2\sqrt{2}}{3-2\sqrt{2}}$

$\implies \frac{a}{b} = \frac{3+2\sqrt{2}}{{3-2\sqrt{2}}}$

$\therefore \boxed{\bold{a:b=(3+2\sqrt{2}): (3-2\sqrt{2})}}$