Write the equation of the circle centered at \( (1,-5) \) with radius 5 . Question Help: Video \( \square \) Message instructor Submit Question Jump to Answer

The equation of a circle is defined by the formula \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius. Here, with the center at \( (1, -5) \) and a radius of \( 5 \), we can plug in these values into the formula. This gives us \( (x - 1)^2 + (y + 5)^2 = 25 \). Circles are not just pretty shapes; they have fascinating properties! For instance, every point on the circle is equidistant from the center, making them great for understanding concepts like symmetry and distance in math. This is why circles are often used in engineering and architecture for designs that need a lot of stability and aesthetic beauty. Did you know that circles have a rich history in mathematics? Ancient civilizations, such as the Egyptians and Greeks, were fascinated by circles, leading to concepts such as pi (\( \pi \)) that are crucial for calculations involving circular shapes. The work of mathematicians like Archimedes laid the groundwork for our modern understanding of geometry and calculus related to circular shapes!