Let the two numbers be a and b.

G.M. = √ab

According to the given condition,

a + b = 6√ab

squaring on both sides we get,

⇒ (a + b)² = 36(ab) —1

Here,

(a—b)² = (a + b)² – 4ab = 36ab – 4ab = 32ab

⇒ a—b = √32√ab = 4√2√ab —2

Adding eq(1) and eq(2), we get

2a = (6 + 4√2)√ab

⇒ a = (3 + 2√2)√ab

Substituting the value of a in eq(1), we obtain

b = 6√ab – (3 + 2√2)√ab

⇒ b = (3 — 2√2)√ab

Now,

∴ The required ratio is