Answer:
See explanation
Step-by-step explanation:
You are given the equality
[tex](x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2[/tex]
where x, y are two positive integers with x>y.
1. x=2,y=1, then
[tex]c=x^2+y^2=2^2+1^2=5\\ \\a=x^2-y^2=2^2-1^2=3\\ \\b=2xy=2\cdot 2\cdot 1=4[/tex]
First Pythagorean triple is (3,4,5)
2. x=3,y=1, then
[tex]c=x^2+y^2=3^2+1^2=10\\ \\a=x^2-y^2=3^2-1^2=8\\ \\b=2xy=2\cdot 3\cdot 1=6[/tex]
Second Pythagorean triple is (6,8,10)
3. x=3,y=2, then
[tex]c=x^2+y^2=3^2+2^2=13\\ \\a=x^2-y^2=3^2-2^2=5\\ \\b=2xy=2\cdot 3\cdot 2=12[/tex]
Third Pythagorean triple is (5,12,13)
4. x=4,y=1, then
[tex]c=x^2+y^2=4^2+1^2=17\\ \\a=x^2-y^2=4^2-1^2=15\\ \\b=2xy=2\cdot 4\cdot 1=8[/tex]
Fourth Pythagorean triple is (8,15,17)
5. x=4,y=2, then
[tex]c=x^2+y^2=4^2+2^2=20\\ \\a=x^2-y^2=4^2-2^2=12\\ \\b=2xy=2\cdot 4\cdot 2=16[/tex]
Fifth Pythagorean triple is (12,16,20)
6. x=4,y=3, then
[tex]c=x^2+y^2=4^2+3^2=25\\ \\a=x^2-y^2=4^2-3^2=7\\ \\b=2xy=2\cdot 4\cdot 3=24[/tex]
Sixth Pythagorean triple is (7,24,25)
7. x=5,y=1, then
[tex]c=x^2+y^2=5^2+1^2=26\\ \\a=x^2-y^2=5^2-1^2=24\\ \\b=2xy=2\cdot 5\cdot 1=10[/tex]
Seventh Pythagorean triple is (10,24,26)
8. x=5,y=2, then
[tex]c=x^2+y^2=5^2+2^2=29\\ \\a=x^2-y^2=5^2-2^2=21\\ \\b=2xy=2\cdot 5\cdot 2=20[/tex]
8th Pythagorean triple is (20,21,29)
9. x=5,y=3, then
[tex]c=x^2+y^2=5^2+3^2=34\\ \\a=x^2-y^2=5^2-3^2=16\\ \\b=2xy=2\cdot 5\cdot 3=30[/tex]
9th Pythagorean triple is (16,30,34)
10. x=5,y=4, then
[tex]c=x^2+y^2=5^2+4^2=41\\ \\a=x^2-y^2=5^2-4^2=9\\ \\b=2xy=2\cdot 5\cdot 4=40[/tex]
10th Pythagorean triple is (9,40,41)
