Thus, the center is \( (3, -2) \) and the radius \( r = \sqrt{25} = 5 \).

["Understanding the Equation of a Circle: Center at (3, −2) and Radius ( r = 5 )", "The equation of a circle is a foundational concept in coordinate geometry, providing a precise way to describe all points at a fixed distance from a central point. A standard circle centered at ( (h, k) ) with radius ( r ) follows the elegant formula:", "[\n(x - h)^2 + (y - k)^2 = r^2\n]", "For the specific circle we’ll explore, the center lies at ( (3, -2) ) and the radius is ( r = \sqrt{25} = 5 ). Substituting these values into the general equation gives:", "[\n(x - 3)^2 + (y + 2)^2 = 25\n]", "This equation defines every point ( (x, y) ) that lies exactly 5 units away from the point ( (3, -2) ) in the Cartesian plane.", "### Why This Equation Matters", "Understanding the circle’s equation is essential in various fields, from geometry and trigonometry to computer graphics and physics. The center ( (3, -2) ) gives the exact midpoint of the circle, while the radius ( r = 5 ) determines how far each point on the circle extends from the center. This clarity makes work with circular motion, symmetry, and distance calculations efficient and accurate.", "### Deriving the Equation Step-by-Step", "To derive this equation, begin with the distance formula between a point ( (x, y) ) and the center ( (3, -2) ):", "[\n\ ext{Distance} = \sqrt{(x - 3)^2 + (y + 2)^2}\n]", "Since all points on the circle are 5 units away, set this equal to the radius ( r = 5 ):", "[\n\sqrt{(x - 3)^2 + (y + 2)^2} = 5\n]", "Squaring both sides removes the square root and yields the standard circle equation:", "[\n(x - 3)^2 + (y + 2)^2 = 25\n]", "### Real-World Applications", "Circles modeled by equations like this appear frequently in real life—think wheels, gears, cameras with circular apertures, or even planetary orbits. Knowing the center and radius allows us to plot, analyze, and simulate these circular systems precisely.", "### Conclusion", "A circle centered at ( (3, -2) ) with radius ( 5 ) is more than just a geometric shape—it’s a powerful mathematical tool. Its equation,\n[\n(x - 3)^2 + (y + 2)^2 = 25\n]\nencapsulates symmetry, distance, and spatial precision. Whether solving math problems or designing engineering systems, mastering this concept is key to unlocking deeper insights into planar geometry.", "---", "Keywords: circle equation center (3, -2) radius 5, (x - 3)² + (y + 2)² = 25, geometry, coordinate geometry, circle properties, math tutorial, coordinate plane, define a circle equation."]