If a : b = c : d , then prove that a 2 + c 2 b 2 + d 2 = a c b d

Given, a : b = c : d ⇒ a b = c d Let a b = c d = k Then, a = b k and c = d k We also given that a 2 + c 2 b 2 + d 2 = a c b d Taking L . H . S . = a 2 + c 2 b 2 + d 2 = b k 2 + d k 2 b 2 + d 2 = b 2 k 2 + d 2 k 2 b 2 + d 2 = k 2 b 2 + d 2 b 2 + d 2 = k 2 . . . . . . . . . 1 R . H . S . = a c b d = b k d k b d = k 2 b d b d = k 2 . . . . . . . . . . . . . 2 From equation 1 and 2 , we get a 2 + c 2 b 2 + d 2 = a c b d Hence proved.

If 8 : 14 : : x : 28 , then find the value of x .

Given: 8 : 14 : : x : 28 We know that, Product of means = Product of extremes 14 × x = 8 × 28 x = 8 × 28 14 x = 224 14 = 16 Hence, the value of x is 16 .

If x + 3 : 27 : : 1 : 3 , then find the value of x .

Given: x + 3 : 27 : : 1 : 3 We know that, Product of means = Product of extremes 27 × 1 = x + 3 × 3 27 = 3 x + 9 3 x = 18 x = 18 3 = 6 Hence, the value of x is 6 .