The problem provides a circle centered at the origin with radius 5 and asks for its equation.
Recall the general equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 .
Substitute the given center (0, 0) and radius 5 into the equation: x 2 + y 2 = 5 2 .
The equation of the circle is x 2 + y 2 = 5 2 .
Explanation
Problem Analysis The problem states that we have a circle centered at the origin (0, 0) with a radius of 5. We need to find the equation of this circle from the given options.
Recall the general equation of a circle The general equation of a circle with center (h, k) and radius r is given by: ( x − h ) 2 + ( y − k ) 2 = r 2
Substitute the given values Since the circle is centered at the origin, we have h = 0 and k = 0. The radius is given as r = 5. Substituting these values into the general equation, we get: ( x − 0 ) 2 + ( y − 0 ) 2 = 5 2 x 2 + y 2 = 25
Compare with the given options Comparing this equation with the given options, we see that option A matches our result: x 2 + y 2 = 5 2
State the final answer Therefore, the equation of the circle is x 2 + y 2 = 5 2 .
Examples
Circles are fundamental in many real-world applications. For instance, understanding the equation of a circle is crucial in designing circular gears in machinery, mapping the coverage area of a Wi-Fi router, or even planning the layout of a circular garden. In architecture, circular designs often require precise mathematical equations to ensure structural integrity and aesthetic appeal. Knowing the equation of a circle allows engineers and designers to accurately model and construct these circular elements.