If the center of a circle is at (0, 0) and it passes through the point (3, 4), what is the equation of the circle?

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Q: If the center of a circle is at (0, 0) and it passes through the point (3, 4), what is the equation of the circle?

Solution: The radius is 5 (distance from (0,0) to (3,4)), so the equation is x² + y² = 5² = 25.

Steps: 9

Step 1: Identify the center of the circle, which is given as (0, 0).
Step 2: Identify a point on the circle, which is given as (3, 4).
Step 3: Calculate the radius of the circle using the distance formula. The distance formula is: distance = √((x2 - x1)² + (y2 - y1)²).
Step 4: Plug in the coordinates of the center (0, 0) and the point (3, 4) into the distance formula: distance = √((3 - 0)² + (4 - 0)²).
Step 5: Simplify the equation: distance = √(3² + 4²) = √(9 + 16) = √25.
Step 6: Calculate the square root: √25 = 5. This means the radius of the circle is 5.
Step 7: Write the equation of the circle using the standard form: (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
Step 8: Substitute the center (0, 0) and the radius 5 into the equation: (x - 0)² + (y - 0)² = 5².
Step 9: Simplify the equation: x² + y² = 25.