What is the radius of a circle whose equation is $x^2+y^2-10 x+6 y+18=0$?
A. 8 units
B. 4 units
C. 2 units
D. 16 units
Answer (2)
Rewrite the given equation x 2 + y 2 − 10 x + 6 y + 18 = 0 by completing the square for both x and y terms. This leads to the equation ( x − 5 ) 2 + ( y + 3 ) 2 = 16 .
Recognize that the equation is now in the standard form of a circle ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
Identify that r 2 = 16 by comparing the derived equation with the standard form.
Calculate the radius by taking the square root of 16, which gives r = 4 . Therefore, the radius of the circle is 4 .
Explanation
Analyze the problem and rewrite the equation We are given the equation of a circle: x 2 + y 2 − 10 x + 6 y + 18 = 0 . Our goal is to find the radius of this circle. To do this, we will rewrite the equation in the standard form of a circle's equation, which is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center of the circle and r is the radius.
Complete the square for x terms First, let's complete the square for the x terms. We have x 2 − 10 x . To complete the square, we need to add and subtract ( 2 10 ) 2 = 5 2 = 25 . So, x 2 − 10 x = ( x − 5 ) 2 − 25 .
Complete the square for y terms Next, let's complete the square for the y terms. We have y 2 + 6 y . To complete the square, we need to add and subtract ( 2 6 ) 2 = 3 2 = 9 . So, y 2 + 6 y = ( y + 3 ) 2 − 9 .
Substitute back into the original equation Now, substitute these back into the original equation: ( x − 5 ) 2 − 25 + ( y + 3 ) 2 − 9 + 18 = 0 .
Simplify the equation Simplify the equation: ( x − 5 ) 2 + ( y + 3 ) 2 = 25 + 9 − 18 = 16 .
Identify the radius Now the equation is in the standard form ( x − 5 ) 2 + ( y + 3 ) 2 = 16 . Comparing this to ( x − h ) 2 + ( y − k ) 2 = r 2 , we see that r 2 = 16 .
Find the radius Therefore, the radius is r = 16 = 4 . The radius of the circle is 4 units.
Examples
Understanding the radius of a circle is crucial in many real-world applications. For example, when designing a circular garden, the radius determines the amount of fencing needed. If you want to build a circular garden with the equation x 2 + y 2 − 10 x + 6 y + 18 = 0 , you now know the radius is 4 units, which helps you determine the area you'll be planting in and the amount of materials you'll need. Another example is in architecture, where circular windows or domes require precise radius calculations for structural integrity and aesthetic appeal.
The radius of the circle given by the equation x 2 + y 2 − 10 x + 6 y + 18 = 0 is calculated to be 4 units by completing the square and rewriting the equation in standard form. Therefore, the answer is B .4 units.
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