Q: If roots of the equation x^{2} – 10 c x – 11 d = 0 are a, b and those of x^{2} – 10 a x – 11 b = 0 are c , d , then find the value of a + b + c + d. (a, b, c and d are distinct numbers)

Solution : As a + b = 10 c and c + d = 10 a

ab = -11 d , cd = -11 b

⇒ ac = 121 and (b + d) = 9(a + c)

a^{2} – 10 ac – 11d = 0

c^{2} – 10 ac – 11b = 0

⇒ a^{2} + c^{2} – 20ac – 11(b + d) = 0

⇒ (a + c)^{2} – 22(121) – 11 × 9(a + c) = 0

⇒ (a + c) = 121 or -22 (rejected)

Hence , a + b + c + d = 1210.