Points on circle

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

Final Answer:

n =  12

Step-by-step explanation:

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

The equation has the following integer solutions:

x^2 + y^2 = 25

Number of solutions found: 14

x₁=-5; y₁=0
x₂=-5; y₂=-0
x₃=-4; y₃=3
x₄=-4; y₄=-3
x₅=-3; y₅=4
x₆=-3; y₆=-4
x₇=0; y₇=5
x₈=0; y₈=-5
x₉=3; y₉=4
x₁₀=3; y₁₀=-4
x₁₁=4; y₁₁=3
x₁₂=4; y₁₂=-3
x₁₃=5; y₁₃=0
x₁₄=5; y₁₄=-0

Calculated by our Diofant problems and integer equations.

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