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. By rewriting the given equation ( x + 9 ) 2 + ( y − 6 ) 2 = 1 0 2 in the standard form, we identify the center as follows:
Rewrite the equation: ( x − ( − 9 ) ) 2 + ( y − 6 ) 2 = 1 0 2 .
Identify the center: ( h , k ) = ( − 9 , 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, our equation becomes ( x − ( − 9 ) ) 2 + ( y − 6 ) 2 = 1 0 2 .
Identifying the Center By comparing this with the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , we can identify the coordinates of the center as 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 example, in navigation, GPS systems use circles to determine the location of a device. If a GPS satellite sends a signal indicating you are within a certain radius of its location, your device calculates its possible locations along that circle. The intersection of circles from multiple satellites narrows down your precise location. This principle relies on the fundamental equation of a circle and its center.
