Circle $Q$ is centered at the origin and has a radius of 9 units. Which equation could represent circle Q?
A. $x^2+y^2=3$
B. $x^2+y^2=9$
C. $x^2+y^2=81$
D. $(x-9)^2+(y-9)^2=1$
Answer (1)
The 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 is 9, substitute r = 9 into the equation: x 2 + y 2 = 9 2 = 81 .
The equation of circle Q is x 2 + y 2 = 81 .
Explanation
Analyze the problem The problem states that circle Q is centered at the origin and has a radius of 9 units. We need to find the equation that represents this circle.
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 Q is centered at the origin, we have h = 0 and k = 0 . The radius is given as r = 9 . Substituting these values into the general equation, we get: ( x − 0 ) 2 + ( y − 0 ) 2 = 9 2 x 2 + y 2 = 81
State the final equation Therefore, the equation that represents circle Q is x 2 + y 2 = 81 .
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, when designing a circular garden, knowing the radius and center allows you to determine the exact layout using the circle's equation. Similarly, in computer graphics, circles are fundamental shapes, and their equations are used to draw and manipulate circular objects on the screen. This knowledge also extends to fields like astronomy, where the orbits of planets and satellites can be approximated as circles or ellipses, and their positions can be described using mathematical equations.
