The average of two numbers is 6.5 and square root of their product is 6. What are the numbers?

Answer: Option C

Let the two numbers be x and y
Then,
x + y = 6.5 × 2 = 13 and
$$\sqrt{\text{xy}}$$  = 6 or xy = 36
⇒ (x - y)² = (x + y)² - 4xy
⇒ (x - y)² = (13)² - 4 × 36
⇒ (x - y)² = 169 - 144
⇒ (x - y)² = 25
⇒ (x - y) = 5
Solving x + y = 13 and x - y = 5
We get : x = 9 , y = 4