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 ( x + 9 ) 2 = ( x − ( − 9 ) ) 2 , we find h = − 9 .
From ( y − 6 ) 2 , we find k = 6 . Therefore, the center 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 . Similarly, ( y − 6 ) 2 directly gives us k = 6 . Therefore, the center of the circle is ( − 9 , 6 ) .
State the final answer The center of the circle is ( − 9 , 6 ) .
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, when designing a circular garden, knowing the center and radius helps in accurately planning the layout and ensuring the garden fits perfectly within the available space. Similarly, in navigation, the equation of a circle can represent the range of a radio signal, with the center indicating the location of the transmitter and the radius representing the signal's reach. This allows for precise positioning and coverage planning.
