Write the equation of the circle centered at \( (-8,6) \) with diameter 16 . Question Help: Video \( \square \) Message instructor Submit Question Jump to Answer

The equation of a circle is given by \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center of the circle and \( r \) is the radius. Since the center is \( (-8,6) \) and the diameter is 16, the radius \( r \) will be half of that, which is 8. Therefore, the equation becomes: \[ (x + 8)^2 + (y - 6)^2 = 8^2 \] or \[ (x + 8)^2 + (y - 6)^2 = 64. \] Circles have been studied since ancient times, first by Greeks who found their perfect shapes fascinating. They reflected harmony and beauty in art and architecture. Fast forward to today, the circle serves a vital role in various fields, from physics using circular motion concepts to computer graphics utilizing circular objects for design. It's fascinating how this simple shape has transcended time and discipline! When working with the circle's equation, a common mistake is confusing the diameter with the radius. Remember, the radius is always half of the diameter! Also, while plotting, ensure you center your coordinate axes correctly to avoid skewing the circle's true placement. Keep those coordinates in check to show off that perfect round shape!