To determine which of the given points lies on the circle described by the equation \((x-2)^2+(y+3)^2=4\), we'll check each point one by one to see if it satisfies the circle equation.
### Step-by-step Verification
1. Checking the point \((1, -4)\):
- Substitute \(x = 1\) and \(y = -4\) into the equation:
[tex]\[
(1 - 2)^2 + (-4 + 3)^2 = 4
\][/tex]
- Simplify each part:
[tex]\[
(-1)^2 + (-1)^2 = 1 + 1 = 2
\][/tex]
- This does not equal 4, so the point \((1, -4)\) is not on the circle.
2. Checking the point \((2, 0)\):
- Substitute \(x = 2\) and \(y = 0\) into the equation:
[tex]\[
(2 - 2)^2 + (0 + 3)^2 = 4
\][/tex]
- Simplify each part:
[tex]\[
(0)^2 + (3)^2 = 0 + 9 = 9
\][/tex]
- This does not equal 4, so the point \((2, 0)\) is not on the circle.
3. Checking the point \((0, 0)\):
- Substitute \(x = 0\) and \(y = 0\) into the equation:
[tex]\[
(0 - 2)^2 + (0 + 3)^2 = 4
\][/tex]
- Simplify each part:
[tex]\[
(-2)^2 + (3)^2 = 4 + 9 = 13
\][/tex]
- This does not equal 4, so the point \((0, 0)\) is not on the circle.
4. Checking the point \((2, -5)\):
- Substitute \(x = 2\) and \(y = -5\) into the equation:
[tex]\[
(2 - 2)^2 + (-5 + 3)^2 = 4
\][/tex]
- Simplify each part:
[tex]\[
(0)^2 + (-2)^2 = 0 + 4 = 4
\][/tex]
- This does equal 4, so the point \((2, -5)\) is on the circle.
### Conclusion
After checking each of the given points, we find that the point [tex]\((2, -5)\)[/tex] lies on the circle described by the equation [tex]\((x-2)^2+(y+3)^2=4\)[/tex].
