Question Find the center and radius of the circle represented by the equation below. \[ x^{2}+y^{2}-10 x-14 y+38=0 \] Answer Attempt i out of 2 Center: \( ( \) Radius: Submit Answer Vid

To find the center and radius of the circle from the equation \( x^{2}+y^{2}-10 x-14 y+38=0 \), we first rewrite it in standard form by completing the square. 1. Rearranging gives us: \[ x^{2} - 10 x + y^{2} - 14 y = -38 \] 2. Completing the square for \( x \): \[ x^{2} - 10 x = (x - 5)^{2} - 25 \] 3. Completing the square for \( y \): \[ y^{2} - 14 y = (y - 7)^{2} - 49 \] 4. Substituting back, we have: \[ (x - 5)^{2} - 25 + (y - 7)^{2} - 49 = -38 \] \[ (x - 5)^{2} + (y - 7)^{2} = 36 \] Thus, the center is \( (5, 7) \) and the radius is \( \sqrt{36} = 6 \). Center: \( (5, 7) \) Radius: \( 6 \) Now, let’s dive into some history and practical applications of circles! In ancient Greece, mathematicians like Euclid and Archimedes studied circles extensively, laying down principles that govern not just geometry but also astronomy and the study of orbits. Their explorations helped pave the way for our modern understanding of circular shapes, paving the way for later innovations in engineering and architecture. Circles are not just mathematical concepts; they have real-world applications in fields like engineering and technology! For instance, when designing wheels, gears, or even in the construction of bridges, understanding the properties of circles can lead to efficient and robust designs. Whether it's ensuring that a roller coaster has the correct curve or that a roundabout facilitates smooth traffic flow, circles are crucial in countless everyday phenomena!