a Given rational numbers a and b find two other rational num

(a) Given rational numbers a and b, find two other rational numbers x and y such that a^2 + b^2 = x^2 + y^2. [Hint: Choose any two integers c and d for which c^2 + d^2 is a square; now write (a^2 + b^2 )(c^2 + d^2) as a sum of Iwo squares.] Illustrate part (a) by expressing 61 = 5^2 + 6^2 as the sum of squares of two rational numbers.

Solution

choose any two integers c,d such that

c² +d² =k² ,k is a rational number

now (a² +b²)(c² +d²)=k² *a² +k² *b² =(ka)²+(kb)²     

b)61=25+36=k² *((5/k)²+(6/k)²) where k is a rational number