Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.
Let the two natural numbers be x and y.
According to the question
x2 + y2 = 25(x + y) – – – – – (1)
x2 + y2 = 50(x – y) – – – – (2)
From (1) and (2) we get
25(x + y) = 50(x – y)
x + y = 2(x – y)
x + y = 2x – 2y
y + 2y = 2x – x
3y = x – – – – – (3)
From (2) and (3) we get
(3y)2 + y2 = 50(3y – y)
9y2 + y2 = 50(3y – y)
10 y2 = 100y
y = 10
From (3) we have,
x = 3y = 3.10 = 30
Hence the two natural numbers are 30 and 10.