Identify the radius and the center of a circle whose equation is $(x-5)^2+y^2=81$.
The radius of the circle is $\square$ units.
The center of the circle is at ( $\square$ , $\square$ ).
Answer (2)
The equation of the circle is in the form ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
By comparing the given equation ( x − 5 ) 2 + y 2 = 81 with the general form, we identify h = 5 and k = 0 .
We find the radius by taking the square root of the constant term: r = 81 = 9 .
The center of the circle is ( 5 , 0 ) and the radius is 9 .
Explanation
Analyze the problem and given data. We are given the equation of a circle as ( x − 5 ) 2 + y 2 = 81 . Our goal is to identify the radius and the center of this circle. We know that the general equation of a circle is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) represents the center of the circle and r is the radius.
Identify the center and radius. Comparing the given equation ( x − 5 ) 2 + y 2 = 81 with the general equation ( x − h ) 2 + ( y − k ) 2 = r 2 , we can identify the values of h , k , and r .
For the x-coordinate of the center, we have ( x − 5 ) 2 which corresponds to ( x − h ) 2 . Therefore, h = 5 .
For the y-coordinate of the center, we have y 2 , which can be written as ( y − 0 ) 2 . This corresponds to ( y − k ) 2 , so k = 0 .
For the radius, we have r 2 = 81 . To find r , we take the square root of 81: r = 81 = 9
State the center and radius. Thus, the center of the circle is ( h , k ) = ( 5 , 0 ) and the radius is r = 9 .
Final Answer. The radius of the circle is 9 units, and the center of the circle is at (5, 0).
Examples
Understanding the equation of a circle is useful in various real-world applications. For example, when designing a circular garden, you need to know the center and radius to plan the layout accurately. Similarly, in architecture or engineering, knowing the equation of a circular structure helps in precise construction and design. Also, in computer graphics, circles are fundamental shapes, and their equations are used to draw and manipulate them on the screen. For instance, if you're creating a game where a character moves in a circle around a central point, you'd use the circle's equation to calculate the character's position at any given time.
The radius of the circle is 9 units, and the center of the circle is at (5, 0).
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