Select the correct answer from each drop-down menu.
Rose graphs this equation as shown:
[tex]$x^2-4 x+y^2=0$
[/tex]
Rose correctly found the $\square$ but incorrectly found the $\square$. To correct her graph, she should $\square$.
Answer (2)
Rose correctly identified the radius of the circle as 2 but misidentified the center as (0,0). The correct center should be (2,0). To fix the graph, she should shift the center to (2,0) while keeping the radius as 2.
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Rewrite the given equation x 2 − 4 x + y 2 = 0 by completing the square to get ( x − 2 ) 2 + y 2 = 4 .
Identify the center of the circle as ( 2 , 0 ) and the radius as 2 .
Recognize that Rose correctly found the radius but incorrectly found the center.
To correct the graph, Rose should shift the center from ( 0 , 0 ) to ( 2 , 0 ) while keeping the radius as 2, so the final answer is: r a d i u s .
Explanation
Analyze the equation Let's analyze the given equation and determine the correct parameters for the circle. The equation is:
x 2 − 4 x + y 2 = 0
To find the standard form of the circle equation, we need to complete the square.
Complete the square Completing the square for the x terms:
x 2 − 4 x can be rewritten as ( x − 2 ) 2 − 4 .
So, the equation becomes:
( x − 2 ) 2 − 4 + y 2 = 0
( x − 2 ) 2 + y 2 = 4
Identify center and radius Now we can identify the center and radius of the circle. The standard form of a circle equation is:
( x − h ) 2 + ( y − k ) 2 = r 2
where ( h , k ) is the center and r is the radius.
Comparing this with our equation ( x − 2 ) 2 + y 2 = 4 , we have:
Center: ( 2 , 0 ) Radius: 4 = 2
Determine what Rose did correctly and incorrectly Now, let's consider what Rose might have done correctly and incorrectly. Since the equation is x 2 − 4 x + y 2 = 0 , it's possible she correctly identified the radius as 2. However, she might have incorrectly determined the center. The correct center is ( 2 , 0 ) , not ( 0 , 0 ) .
To correct her graph, she should shift the center of the circle from ( 0 , 0 ) to ( 2 , 0 ) while keeping the radius as 2.
Examples
Understanding circle equations is crucial in various fields. For example, in navigation, GPS systems use coordinates to pinpoint locations on a map, which can be modeled using circle equations. Imagine you're designing a circular garden with a pond at the center. By knowing the equation of the circle, you can easily determine the garden's boundaries and plan the layout effectively. The equation ( x − 2 ) 2 + y 2 = 4 represents a garden with a center 2 units to the right of the origin and a radius of 2 units.
