The equation of a circle \( x(x-10)+y(y-4)=24 \) find the center and radius of the circle. Working

To rewrite the equation of the circle, expand it first: \[ x^2 - 10x + y^2 - 4y = 24 \] Now, we want to bring the equation in standard form \((x - h)^2 + (y - k)^2 = r^2\) by completing the square for both the \(x\) and \(y\) terms. For \(x\): 1. Take \(-10\), halve it to get \(-5\), and square it to get \(25\). 2. Rewrite: \[ x^2 - 10x = (x - 5)^2 - 25 \] For \(y\): 1. Take \(-4\), halve it to get \(-2\), and square it to get \(4\). 2. Rewrite: \[ y^2 - 4y = (y - 2)^2 - 4 \] Substituting back into the equation gives: \[ (x - 5)^2 - 25 + (y - 2)^2 - 4 = 24 \] Combine like terms: \[ (x - 5)^2 + (y - 2)^2 - 29 = 24 \] Rearranging provides: \[ (x - 5)^2 + (y - 2)^2 = 53 \] From the standard form, we can identify the center and radius. The center \((h, k)\) is \((5, 2)\) and the radius \(r\) is \(\sqrt{53}\). In conclusion, the center of the circle is \((5, 2)\) and the radius is \(\sqrt{53}\). Now, let’s sprinkle in some engaging tidbits! Did you know that circles were studied as far back as ancient Greece? Mathematicians like Euclid laid the groundwork for geometry, including the study of circles, defining their properties and relationships long before calculators were available! Circles represent perfectly symmetrical shapes, making them a favorite in everything from art to architecture. Moreover, understanding the equation of a circle isn't just a fun math exercise; it's crucial in various fields! From computer graphics, where circles help create smooth curves and shapes, to physics, where they model rotational motion, mastery of circle equations can lead to exciting real-world applications! Let that circle roll into your knowledge!