Exactly ? different pairs of integers (x,y) represent points on the circle x^2 + y^2 = 25.
A. 12
B. 8
C. 6
D. 4
To find the number of different pairs of integers (x,y) that represent points on the circle x^2 + y^2 = 25, we can use the following approach:
Step 1: List all possible integer values for x and y within a certain range.
Since the equation is x^2 + y^2 = 25, we know that the maximum absolute value of x or y can be 5, since 5^2 = 25. Therefore, we can list all possible integer values for x and y by considering all integers from -5 to 5.
Step 2: Substitute the values of x and y into the equation x^2 + y^2 = 25.
For each pair of values (x,y), substitute them into the equation to check if they satisfy the equation x^2 + y^2 = 25. If the equation is true, then the pair (x,y) represents a point on the circle.
Step 3: Count the number of valid pairs.
Count the number of pairs that satisfy the equation x^2 + y^2 = 25. Each valid pair represents a unique point on the circle.
By following this approach, we can determine the number of different pairs of integers (x,y) that represent points on the circle x^2 + y^2 = 25.
Now let's go through the steps:
Step 1: List all possible integer values for x and y.
The possible integer values for x and y are: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, and 5.
Step 2: Substitute the values of x and y into the equation.
Substituting the values of x and y into the equation x^2 + y^2 = 25, we get:
(-5)^2 + (-4)^2 = 25 (valid)
(-5)^2 + (-3)^2 = 34 (not valid)
(-5)^2 + (-2)^2 = 29 (not valid)
(-5)^2 + (-1)^2 = 26 (not valid)
(-5)^2 + (0)^2 = 25 (valid)
(-5)^2 + (1)^2 = 26 (not valid)
...
(5)^2 + (3)^2 = 34 (not valid)
(5)^2 + (4)^2 = 41 (not valid)
(5)^2 + (5)^2 = 50 (not valid)
Step 3: Count the number of valid pairs.
By counting the number of valid pairs, we find that there are 4 different pairs of integers (x,y) that represent points on the circle x^2 + y^2 = 25.
Therefore, the correct answer is option D. 4.