Give the equation of the circk centered at the crigin and passing through the point \( (0,-9) \),

The equation of a circle is given by the formula \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center of the circle and \( r \) is the radius. Since the circle is centered at the origin \( (0, 0) \), the equation simplifies to \( x^2 + y^2 = r^2 \). To find \( r \), we calculate the distance from the origin to the point \( (0, -9) \), which is 9. Thus, the equation of the circle is \( x^2 + y^2 = 81 \). Did you know that the concept of the circle dates back to ancient civilizations? The Greeks, especially, were fascinated by circles, linking them to perfection and harmony in mathematics. The famous mathematician Euclid even wrote about circles and their properties in his work "Elements," laying the groundwork for geometry as we know it today! When solving problems involving circles, one common mistake is forgetting to consider whether the point given lies on the circle or not. Always double-check your work by plugging the coordinates of the point into the equation. If the left and right sides of the equation balance, then your point is indeed on the circle! Remember, circles can sometimes be a bit tricky with their radii and centers, but practice makes perfect!