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 the circle is ( x + 9 ) 2 + ( y − 6 ) 2 = 1 0 2 .
The general form of a circle's equation is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center.
Comparing the given equation with the general form, we identify h = − 9 and k = 6 .
The center of the circle is ( − 9 , 6 ) .
Explanation
Problem Analysis The equation of a circle is given by ( x + 9 ) 2 + ( y − 6 ) 2 = 1 0 2 . We need to find the center of this circle.
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 . Comparing this with the given equation ( x + 9 ) 2 + ( y − 6 ) 2 = 1 0 2 , we can identify the values of h and k .
Finding the Center We can rewrite the given equation as ( x − ( − 9 ) ) 2 + ( y − 6 ) 2 = 1 0 2 . By comparing this with the general form ( x − h ) 2 + ( y − k ) 2 = r 2 , we can see that h = − 9 and k = 6 . Therefore, the center of the circle is ( − 9 , 6 ) .
Final Answer The center of the circle is ( − 9 , 6 ) .
Examples
Understanding the equation of a circle is crucial in various fields, such as engineering and computer graphics. For instance, when designing a circular garden, knowing the center and radius helps in accurately plotting the layout. Similarly, in computer graphics, defining circles using their center coordinates and radius is fundamental for creating circular shapes and animations. This concept also extends to physics, where circular motion is analyzed using the circle's equation to determine parameters like angular velocity and centripetal force.
