What is the equation of a circle centered at the origin with radius [tex]$13$[/tex]?
A. [tex]$x^3+y^3=13$[/tex]
B. [tex]$x^2+y^2=13$[/tex]
C. [tex]$x^3+y^3=169$[/tex]
D. [tex]$x^2+y^2=169$[/tex]
Answer (1)
The general equation of a circle centered at ( h , k ) with radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 .
Since the circle is centered at the origin ( 0 , 0 ) , the equation simplifies to x 2 + y 2 = r 2 .
Given the radius r = 13 , substitute it into the equation: x 2 + y 2 = 1 3 2 .
The equation of the circle is x 2 + y 2 = 169 .
Explanation
Understanding the Circle Equation The equation of a circle centered at ( h , k ) with radius r is given by ( x − h ) 2 + ( y − k ) 2 = r 2 . In this problem, the circle is centered at the origin, which means ( h , k ) = ( 0 , 0 ) , and the radius is r = 13 .
Substituting Values Substitute the values of h , k , and r into the circle equation: ( x − 0 ) 2 + ( y − 0 ) 2 = 1 3 2 Simplify the equation: x 2 + y 2 = 169
Final Answer The equation of the circle centered at the origin with radius 13 is x 2 + y 2 = 169 . Therefore, the correct answer is D.
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, in GPS navigation, your location is determined by the intersection of circles from multiple satellites. The equation of a circle also helps in designing circular structures like domes and amphitheaters, ensuring structural integrity and optimal acoustics. Moreover, in computer graphics, circles are fundamental for creating smooth curves and shapes, enhancing visual appeal and realism.
