Which equation represents the circle described?
* The radius is 2 units.
* The center is the same as the center of a circle whose equation is [tex]x^2+y^2-8 x-6 y+24=0[/tex].
A. [tex](x+4)^2+(y+3)^2=2[/tex]
B. [tex](x-4)^2+(y-3)^2=2[/tex]
C. [tex](x-4)^2+(y-3)^2=2^2[/tex]
D. [tex](x+4)^2+(y+3)^2=2^2[/tex]
Answer (1)
We are given the radius of the circle is 2 and the center is the same as the center of the circle x 2 + y 2 − 8 x − 6 y + 24 = 0 .
We complete the square to find the center of the given circle: ( x 2 − 8 x ) + ( y 2 − 6 y ) = − 24 ⇒ ( x 2 − 8 x + 16 ) + ( y 2 − 6 y + 9 ) = − 24 + 16 + 9 ⇒ ( x − 4 ) 2 + ( y − 3 ) 2 = 1 . Thus, the center is ( 4 , 3 ) .
Using the standard equation of a circle ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius, we have ( x − 4 ) 2 + ( y − 3 ) 2 = 2 2 .
Therefore, the equation of the circle is ( x − 4 ) 2 + ( y − 3 ) 2 = 2 2 .
Explanation
Problem Analysis We are given that the radius of the circle is 2 units and its center is the same as the center of the circle defined by the equation x 2 + y 2 − 8 x − 6 y + 24 = 0 . Our goal is to find the equation of this circle.
Isolating x and y terms To find the center of the circle x 2 + y 2 − 8 x − 6 y + 24 = 0 , we need to complete the square. First, we rewrite the equation as ( x 2 − 8 x ) + ( y 2 − 6 y ) = − 24 .
Completing the square Now, we complete the square for the x terms. We take half of the coefficient of the x term, which is − 8 , so half of that is − 4 . Squaring − 4 gives us 16 . So we add 16 to complete the square for the x terms: ( x 2 − 8 x + 16 ) . Similarly, for the y terms, we take half of the coefficient of the y term, which is − 6 , so half of that is − 3 . Squaring − 3 gives us 9 . So we add 9 to complete the square for the y terms: ( y 2 − 6 y + 9 ) .
Adding constants to both sides We add 16 and 9 to both sides of the equation: ( x 2 − 8 x + 16 ) + ( y 2 − 6 y + 9 ) = − 24 + 16 + 9 . This simplifies to ( x − 4 ) 2 + ( y − 3 ) 2 = 1 .
Finding the center From the equation ( x − 4 ) 2 + ( y − 3 ) 2 = 1 , we can see that the center of the circle is ( 4 , 3 ) .
Finding the equation of the circle The equation of a circle with center ( h , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 . Since the center of our circle is ( 4 , 3 ) and the radius is 2 , the equation of the circle is ( x − 4 ) 2 + ( y − 3 ) 2 = 2 2 .
Final Answer Therefore, the equation of the circle is ( x − 4 ) 2 + ( y − 3 ) 2 = 4 .
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 determine the placement of plants and the overall layout. Similarly, in architecture, circular arches and domes rely on precise calculations of the circle's equation to ensure structural integrity and aesthetic appeal. This knowledge also extends to fields like computer graphics, where circles are fundamental elements in creating images and animations.
