IV. Find the equation of the circle given the center and radius: Given: Center: \( (8,0), r=1 \)

The equation of a circle is derived from its standard form, which is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Substituting the given center \((8, 0)\) and radius \(1\) into the equation gives us: \((x - 8)^2 + (y - 0)^2 = 1^2\). Simplifying this, the equation is \((x - 8)^2 + y^2 = 1\). Circles are fascinating shapes with a rich history, dating back to ancient civilizations! The study of circles dates back to the Babylonians and Egyptians, who explored their properties long before the advent of calculus. The Greeks, particularly Apollonius, even classified conic sections, showing just how much circles piqued the curiosity of mathematicians throughout time. In real-world applications, understanding the equations of circles is crucial in fields like architecture and engineering, where circular structures and designs are prevalent. Whether it's designing domes or roundabouts, circles help create functional and aesthetically pleasing elements in urban designs. This knowledge can also extend to technology, such as in game design for creating circular areas of influence or movement!