Find the center and radius of this circle.
[tex]\begin{array}{l}
(x-2)^2+(y+3)^2=36 \\
C=([?],[]) \text { and } r=[]
\end{array}[/tex]
Answer (1)
Analyze the given circle equation ( x − 2 ) 2 + ( y + 3 ) 2 = 36 and compare it to the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 .
Identify the center's coordinates by recognizing h = 2 and k = − 3 .
Calculate the radius by taking the square root of the right-hand side: r = 36 = 6 .
State the center and radius: C = ( 2 , − 3 ) and r = 6 .
Explanation
Analyze the equation Let's analyze the equation of the circle to find its center and radius. The general form of a circle's equation is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) represents the center of the circle and r is the radius.
Compare with the general form Now, let's compare the given equation ( x − 2 ) 2 + ( y + 3 ) 2 = 36 with the general form. We can see that:
h = 2 (the x-coordinate of the center)
k = − 3 (since y + 3 = y − ( − 3 ) , the y-coordinate of the center)
r 2 = 36
Calculate the radius To find the radius r , we need to take the square root of 36:
r = 36 = 6
So, the radius of the circle is 6.
State the center and radius Therefore, the center of the circle is ( 2 , − 3 ) and the radius is 6.
Examples
Understanding the equation of a circle is very useful in various real-world applications. For example, civil engineers use this concept when designing circular structures such as tunnels or roundabouts. Imagine you're designing a circular roundabout with a radius of 6 meters, centered at the coordinates (2, -3) on a map. The equation ( x − 2 ) 2 + ( y + 3 ) 2 = 36 helps define the exact boundaries of the roundabout, ensuring accurate construction and efficient traffic flow.
