Points on circle

In the Cartesian coordinate system with the origin O is a sketched circle k /O; r=2 cm/. Write all the points that lie on a circle k and whose coordinates are integers. Write all the points that lie on the circle I / O; r=5 cm / and whose coordinates are integers. How many points is it?

Correct result:

n =  12

Solution:

$$\begin{gathered} r_1=2\text{ cm } \\ (x-0{)}^2+(y-0{)}^2=r_1^2 \\ {x}^2+y^2=4 \\ k:[2,0];[-2,0];[0,2];[0,-2] \\ {r}_2=5\text{ cm } \\ (x-0{)}^2+(y-0{)}^2=r_2^2 \\ {x}^2+y^2=25 \\ l:[5,0];[-5,0];[0,5];[0,-5]; \\ [3,4];[3,-4];[-3,4];[-3,-4] \\ [4,3],[-4,3],[4,-3],[-4,-3] \\ n=3\cdot 4=12 \end{gathered}$$

The equation has the following integer solutions:

x^2 + y^2 = 25

Number of solutions found: 12

x₁=-5; y₁=0
x₂=-4; y₂=3
x₃=-4; y₃=-3
x₄=-3; y₄=4
x₅=-3; y₅=-4
x₆=0; y₆=5
x₇=0; y₇=-5
x₈=3; y₈=4
x₉=3; y₉=-4
x₁₀=4; y₁₀=3
x₁₁=4; y₁₁=-3
x₁₂=5; y₁₂=0

Calculated by our Diofant problems and integer equations.

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