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 $x^2+y^2-8 x-6 y+24=0$.
A. $(x+4)^2+(y+3)^2=2$
B. $(x-4)^2+(y-3)^2=2$
C. $(x-4)^2+(y-3)^2=2^2$
D. $(x+4)^2+(y+3)^2=2^2
Answer (2)
Find the center of the circle x 2 + y 2 − 8 x − 6 y + 24 = 0 by completing the square, which gives ( x − 4 ) 2 + ( y − 3 ) 2 = 1 .
Identify the center as ( 4 , 3 ) .
Use the general equation of a circle ( x − h ) 2 + ( y − k ) 2 = r 2 with center ( 4 , 3 ) and radius 2 .
Write the equation of the described circle as ( 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.
Rewrite the Equation 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
Complete the Square Now, we complete the square for the x terms. We take half of the coefficient of the x term, which is − 8 , and square it: ( 2 − 8 ) 2 = ( − 4 ) 2 = 16 . So we add 16 to both sides of the equation within the parenthesis for x terms. Similarly, we complete the square for the y terms. We take half of the coefficient of the y term, which is − 6 , and square it: ( 2 − 6 ) 2 = ( − 3 ) 2 = 9 . So we add 9 to both sides of the equation within the parenthesis for y terms. ( x 2 − 8 x + 16 ) + ( y 2 − 6 y + 9 ) = − 24 + 16 + 9
Simplify the Equation Now we simplify the equation: ( x − 4 ) 2 + ( y − 3 ) 2 = 1
Identify the Center From the simplified equation, we can see that the center of the circle is ( 4 , 3 ) and the radius is 1 = 1 . However, we are looking for a circle with radius 2 and the same center ( 4 , 3 ) . The general equation of a circle with center ( h , k ) and radius r is: ( x − h ) 2 + ( y − k ) 2 = r 2
Write the Equation of the Described Circle In our case, the center is ( 4 , 3 ) and the radius is 2 . Plugging these values into the general equation, we get: ( x − 4 ) 2 + ( y − 3 ) 2 = 2 2
Final Answer Therefore, the equation of the circle with radius 2 and center ( 4 , 3 ) is ( x − 4 ) 2 + ( y − 3 ) 2 = 2 2 .
Examples
Understanding the equation of a circle is crucial in various fields. For instance, in GPS technology, determining your location involves finding the intersection of circles from multiple satellites. Each satellite transmits a signal indicating its distance from your device, which can be represented as a circle's radius. By solving the equations of these circles, your GPS can pinpoint your exact coordinates. Similarly, in medical imaging, the equation of a circle helps in identifying and measuring the size of tumors or other anomalies in scans.
The correct equation representing the circle with a radius of 2 units and a center at (4,3) is ( x − 4 ) 2 + ( y − 3 ) 2 = 2 2 , which corresponds to option C. The center is derived from the original circle equation by completing the square. Thus, the answer is C.
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