PLEASE HELP ASAP!!
Point (4, 2) belongs to the circle with the center at (-1, 1). Which statement verifies that the point (-5, -3) is NOT on the circle?
A. (4 + 1)2 + (2 - 1)2 ≠ (-5 + 1)2 + (-3 - 1)2
B. (4 - 1)2 + (2 + 1)2 ≠ (-5 - 1)2 + (-3 + 1)2
C. (-1 + 1)2 + (4 + 1)2 ≠ (-1 + 1)2 + (-5 - 3)2
D. (-1 - 1)2 + (4 - 1)2 ≠ (-1 - 1)2 + (-5 + 3)2
the general circle equation is ... (x - h)^2 + (y - k)^2 = r^2
(h,k) is the center of the circle , r is the radius
the equations of the two points should NOT be equal
... and they should have the correct format
To verify whether the point (-5, -3) is on the circle with the center at (-1, 1), we can calculate the distance between the center and the given point and then compare it to the radius of the circle.
The distance formula between two points (x1, y1) and (x2, y2) is given by:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
For the given point (4, 2) and the center (-1, 1), the distance is:
d = √[(-5 - (-1))^2 + (-3 - 1)^2] = √[(-5 + 1)^2 + (-3 - 1)^2] = √[(-4)^2 + (-4)^2] = √[16 + 16] = √32
Now, let's evaluate each of the answer choices to see if they hold true:
A. (4 + 1)^2 + (2 - 1)^2 ≠ (-5 + 1)^2 + (-3 - 1)^2
Substituting the values, we have:
25 + 1 ≠ 16 + 16
26 ≠ 32 - This statement is true.
B. (4 - 1)^2 + (2 + 1)^2 ≠ (-5 - 1)^2 + (-3 + 1)^2
Substituting the values, we have:
9 + 9 ≠ 36 + 4
18 ≠ 40 - This statement is true.
C. (-1 + 1)^2 + (4 + 1)^2 ≠ (-1 + 1)^2 + (-5 - 3)^2
Substituting the values, we have:
0 + 25 ≠ 0 + 64
25 ≠ 64 - This statement is true.
D. (-1 - 1)^2 + (4 - 1)^2 ≠ (-1 - 1)^2 + (-5 + 3)^2
Substituting the values, we have:
4 + 9 ≠ 4 + 4
13 ≠ 8 - This statement is false.
From the above analysis, we can see that option D is the only statement that does not hold true. Therefore, the statement that verifies that the point (-5, -3) is NOT on the circle is D.