Questions: Give the equation of the circle centered at the origin and passing through the point (0,4).

Solution Steps

To find the equation of a circle centered at the origin and passing through a given point, we use the standard form of the equation of a circle: \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. Since the circle passes through the point \((0, 4)\), we can substitute these coordinates into the equation to find the radius. Once we have the radius, we can write the full equation of the circle.

To find the equation of the circle centered at the origin and passing through the point \((0, 4)\), we first determine the radius. The radius \(r\) is the distance from the origin \((0, 0)\) to the point \((0, 4)\). Using the distance formula, we have: \[ r = \sqrt{(0 - 0)^2 + (4 - 0)^2} = \sqrt{0 + 16} = \sqrt{16} = 4 \]

The standard form of the equation of a circle centered at the origin is: \[ x^2 + y^2 = r^2 \] Substituting the radius \(r = 4\) into the equation, we get: \[ x^2 + y^2 = 4^2 = 16 \]

Final Answer