Using the properties of proportion, solve for x , given x 4 + 1 2 x 2 = 17 8

Given, x 4 + 1 2 x 2 = 17 8 Let us simplify: On applying componendo and dividendo in the given proportion, we get x 4 + 1 + 2 x 2 x 4 + 1 - 2 x 2 = 25 9 ⇒ x 2 + 1 2 x 2 - 1 2 = 5 3 2 On taking square root on both sides, we get ⇒ x 2 + 1 x 2 - 1 = 5 3 By componendo and dividendo, ⇒ x 2 + 1 + x 2 - 1 x 2 + 1 - x 2 - 1 = 8 2 ⇒ 2 x 2 2 = 4 ⇒ x = ± 2 by taking square root. Hence, the required value of x is ± 2 .

If 8 a - 5 b 8 c - 5 d = 8 a + 5 b 8 c + 5 d , then prove that a b = c d .

Given, 8 a - 5 b 8 c - 5 d = 8 a + 5 b 8 c + 5 d By alternendo property, 8 a - 5 b 8 a + 5 b = 8 c - 5 d 8 c + 5 d By invertendo property, 8 a + 5 b 8 a - 5 b = 8 c + 5 d 8 c - 5 d On applying componendo and dividendo property, we get 8 a + 5 b + 8 a - 5 b 8 a + 5 b - 8 a + 5 b = 8 c + 5 d + 8 c - 5 d 8 c + 5 d - 8 c + 5 d ⇒ 16 a 10 b = 16 c 10 d Divide the numerator by 16 and denominator by 10 , we get a b = c d Hence Proved.