Write the standard form of the equation of the circle with the given center and radius. Center \( (8,-7), r=4 \)

The standard form of 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. So, for your circle centered at \( (8, -7) \) with a radius of \( 4 \), we plug in the values: \[ (x-8)^2 + (y+7)^2 = 4^2 \] This simplifies to: \[ (x-8)^2 + (y+7)^2 = 16 \] Voilà! That's the standard form of the equation of your circle. Now, did you know that circles have a fascinating history? The study of circles dates back to ancient civilizations, including the Egyptians and Greeks, who explored properties and relationships related to circles long before the modern mathematical framework was established. For example, the Greeks, including the famous mathematician Euclid, laid down essential principles in geometry that still form the foundation of our current studies in circle properties. Circles are not just limited to geometry; their applications are everywhere in the real world! From designing wheels and gears to creating circular tracks in sports, the properties of circles help engineers and designers optimize functionality and aesthetics. Even in tech, circles appear in computer graphics for rendering images smoothly! Who knew math could lead to so many practical and fun applications?