Question

Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.

Answer 1

Let the numbers be x and y.
According to the question,

$x^2+y^2=25(x+y)$ ..................(1)

$x^2+y^2=50(x-y)$ ..................(2)

from (1) and (2)

$25(x+y)=50(x-y)$

⇒ $x+y=2(x-y)=2x-2y$

⇒ $y+2y=2x-x$

⇒ $3y=x$

⇒ $x=3y$

putting x in (1)

$x^2+y^2=25(x+y)$

⇒$(3y{)}^2+y^2=25(3y+y)$

⇒$9y^2+y^2=25\times 4y$

⇒$10y^2=100y$

⇒$y^2=10y$

⇒$y=0or10$

but $y\ne 0$ as y is a natural number

So $y=10$

$x=3y=30$

two numbers are 10 and 30