Find the equation of a circle with diameter end points \( (-1,-2) \) and \( (3,2) \) \( \begin{array}{l}\text { [4 mks] } \\ \left.d=\sqrt{\left(-1-(3)^{2}+(-2\right.}-2\right)^{2} \\ d=\sqrt{16+2}\end{array} \)

To find the equation of the circle with the given diameter endpoints, we first calculate the center of the circle, which is the midpoint of the two endpoints. The midpoint \((h, k)\) can be calculated as: \[ h = \frac{-1 + 3}{2} = 1, \quad k = \frac{-2 + 2}{2} = 0. \] So the center of the circle is \((1, 0)\). Next, we find the radius \(r\) by calculating the distance between the center and one of the endpoints. Using the distance formula, we have: \[ r = \sqrt{(1 - (-1))^2 + (0 - (-2))^2} = \sqrt{(1 + 1)^2 + (0 + 2)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}. \] The standard equation of a circle is given by: \[ (x - h)^2 + (y - k)^2 = r^2. \] Substituting \(h = 1\), \(k = 0\), and \(r = 2\sqrt{2}\): \[ (x - 1)^2 + (y - 0)^2 = (2\sqrt{2})^2. \] This simplifies to: \[ (x - 1)^2 + y^2 = 8. \] Therefore, the equation of the circle is: \[ \boxed{(x - 1)^2 + y^2 = 8}. \] Now, let's add some fun background information and applications of circles! A circle has been a part of human history for thousands of years! It’s fascinating to think that the ancient Greeks investigated the properties of circles long before calculators were invented. The simple beauty of a circle—where every point on the perimeter is equidistant from the center—has inspired art, architecture, and mathematics for centuries. Even today, the shape appears in everything from the design of bicycles to the wheels of cars! Circles aren't just mathematical constructs; they have practical applications in real life too! Think about circular tracks in sports, the gears in machines that rotate, or roundabout intersections that help manage traffic flow. Engineers use circles when designing anything from bridges to roller coasters. Plus, they're key in tech—as you move your mouse in a circular motion, you're controlling the movement of your cursor on a screen! So next time you see a circle, remember its rich history and its important role in the world around us!