Final answer:
The equation of a circle with center at (0;0) and a point (3;4) on the perimeter is x² + y² = 25, where the radius is calculated to be 5 using the distance formula.
Step-by-step explanation:
To determine the equation of a circle with center at (0;0) and a point (3;4) on its perimeter, we need to use the standard formula of a circle centered at the origin: x²+ y² = r². First, we calculate the radius (r) of the circle, which is the distance from the center to the point (3;4). We use the distance formula: r = √((3 - 0)² + (4 - 0)²). This simplifies to √(32 + 42) = √(9 + 16) = √25 = 5. Therefore, the radius of our circle is 5. The equation of the circle is thus x² + y² = 52 or x² + y² = 25.