- Define the circle's equation with center \((0,0)\) and radius \(r=5\) as \(x^2 + y^2 = 25\).
- Substitute the coordinates \((x, y)\) of each candidate point into the expression \(x^2 + y^2\).
- Compare the calculated sum to 25.
- Identify the point \((2, \sqrt{21})\) as the one that satisfies the equation.
- Final answer: \(\boxed{(2, \sqrt{21})}\).
Explanation
- Analyze the circle's equation and the condition for a point to lie on it.
A circle centered at the origin \((0, 0)\) with a radius \(r\) has the equation \(x^2 + y^2 = r^2\). Given that the radius is \(r = 5\), we can substitute this value into the equation:
\[x^2 + y^2 = 5^2\]
\[x^2 + y^2 = 25\]
Any point \((x, y)\) that lies on this circle must satisfy this equation. We will test each of the given candidate points to see if their coordinates result in a sum of 25 when squared and added together.
- Test the candidate points.
We test each point by calculating the value of \(x^2 + y^2\):
- For point \((2, \sqrt{21})\):
\[x^2 + y^2 = 2^2 + (\sqrt{21})^2 = 4 + 21 = 25\]
This matches the circle's equation.
- For point \((2, \sqrt{23})\):
\[x^2 + y^2 = 2^2 + (\sqrt{23})^2 = 4 + 23 = 27\]
Since \(27
eq 25\), this point does not lie on the circle.
- For point \((2, 1)\):
\[x^2 + y^2 = 2^2 + 1^2 = 4 + 1 = 5\]
Since \(5
eq 25\), this point does not lie on the circle.
- For point \((2, 3)\):
\[x^2 + y^2 = 2^2 + 3^2 = 4 + 9 = 13\]
Since \(13
eq 25\), this point does not lie on the circle.
- Identify the correct point and conclude.
Based on the calculations performed, only the point \((2, \sqrt{21})\) satisfies the equation \(x^2 + y^2 = 25\). Therefore, this is the only point on the circle.
The final answer is \(\boxed{(2, \sqrt{21})}\).
Examples
Imagine you are programming a simple game where a character moves around a circular path. If the center of the path is at your coordinate origin \((0,0)\) and the radius is 5 units, you can use the equation \(x^2 + y^2 = 25\) to verify if your character's current position \((x, y)\) is exactly on the path. This basic algebraic test ensures that the character stays on the designated circular track, allowing for precise movement control in your game's engine.