What is the center of a circle represented by the equation $(x+9)^2+(y-6)^2=10^2$?
A. $(-9,6)$
B. $(-6,9)$
C. $(6,-9)$
D. $(9,-6)$
Answer (1)
The equation of a circle is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center.
Rewrite the given equation ( x + 9 ) 2 + ( y − 6 ) 2 = 1 0 2 as ( x − ( − 9 ) ) 2 + ( y − 6 ) 2 = 1 0 2 .
Identify the center by comparing with the standard form, which gives h = − 9 and k = 6 .
The center of the circle is ( − 9 , 6 ) .
Explanation
Understanding the Circle Equation The equation of a circle is given by ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) represents the center of the circle and r is the radius. Our given equation is ( x + 9 ) 2 + ( y − 6 ) 2 = 1 0 2 .
Rewriting the Equation We need to rewrite the given equation in the standard form to identify the center. Notice that ( x + 9 ) can be written as ( x − ( − 9 )) . So, we have ( x − ( − 9 ) ) 2 + ( y − 6 ) 2 = 1 0 2 .
Identifying the Center By comparing ( x − ( − 9 ) ) 2 + ( y − 6 ) 2 = 1 0 2 with the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , we can identify the coordinates of the center. We see that h = − 9 and k = 6 . Therefore, the center of the circle is ( − 9 , 6 ) .
Examples
Understanding the equation of a circle is crucial in various fields. For instance, in navigation, GPS systems use circles to determine the range within which a location is situated relative to a satellite. Similarly, in architecture, circular designs are often employed, and knowing the center and radius helps in accurately planning and constructing these structures. In computer graphics, circles are fundamental shapes, and their equations are used extensively to draw and manipulate circular objects on the screen. ( x − h ) 2 + ( y − k ) 2 = r 2
