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 general 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 in the general form to identify h and k .
From the equation, h = − 9 and k = 6 .
The center of the circle is ( − 9 , 6 ) .
Explanation
Identify the general form of a 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 . We need to identify the center ( h , k ) from this equation.
Determine the center of the circle Comparing the given equation ( x + 9 ) 2 + ( y − 6 ) 2 = 1 0 2 with the general form ( x − h ) 2 + ( y − k ) 2 = r 2 , we can rewrite ( x + 9 ) 2 as ( x − ( − 9 ) ) 2 . This tells us that h = − 9 . The term ( y − 6 ) 2 directly gives us k = 6 . Therefore, the center of the circle is ( − 9 , 6 ) .
State the final answer Thus, 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 or a round swimming pool, knowing the center and radius helps in accurately planning the layout and dimensions. In computer graphics, circles are fundamental elements in creating images and animations, where the equation of a circle is used to define and draw circular shapes on the screen.
