This circle is centered at the origin, and the length of its radius is 6. What is the circle's equation?
A. $x^6+y^6=1$
B. $x+y=36$
C. $x^2+y^2=6$
D. $x^2+y^2=36$
Answer (1)
The problem provides a circle centered at the origin with a radius of 6.
The general equation of a circle is ( x − h ) 2 + ( y − k ) 2 = r 2 , where (h, k) is the center and r is the radius.
Substituting the given values (h=0, k=0, r=6) into the general equation gives x 2 + y 2 = 6 2 .
Simplifying, the equation of the circle is x 2 + y 2 = 36 .
Explanation
Problem Analysis The problem states that we have a circle centered at the origin (0, 0) with a radius of 6. 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 = 6. Substituting these values into the general equation, we get: ( x − 0 ) 2 + ( y − 0 ) 2 = 6 2 x 2 + y 2 = 36
Compare with the given options Comparing this equation with the given options, we find that option D matches our derived equation.
State the final answer Therefore, the equation of the circle is x 2 + y 2 = 36 .
Examples
Circles are fundamental in many real-world applications. For example, in architecture, arches and domes often utilize circular geometry for structural integrity and aesthetic appeal. In navigation, understanding circular paths is crucial for determining distances and directions on maps and globes. Moreover, in engineering, circular gears and wheels are essential components in various mechanical systems, showcasing the practical importance of understanding circles and their equations.
