What is the radius of a circle with the equation $x^2+y^2+6 x-2 y+3=0$?
Round your answer to the nearest thousandth.
Answer (1)
Rewrite the given equation x 2 + y 2 + 6 x − 2 y + 3 = 0 in the standard form of a circle ( x − h ) 2 + ( y − k ) 2 = r 2 .
Complete the square for both x and y terms to get ( x + 3 ) 2 + ( y − 1 ) 2 = 7 .
Identify that r 2 = 7 , so the radius is r = 7 .
Round the value of 7 to the nearest thousandth, which is 2.646 . The radius of the circle is 2.646 .
Explanation
Analyze the problem and rewrite the equation We are given the equation of a circle: x 2 + y 2 + 6 x − 2 y + 3 = 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, we complete the square for the x terms. We have x 2 + 6 x . To complete the square, we take half of the coefficient of x , which is 2 6 = 3 , and square it, which is 3 2 = 9 . So, we can rewrite x 2 + 6 x as ( x + 3 ) 2 − 9 .
Complete the square for y terms Next, we complete the square for the y terms. We have y 2 − 2 y . To complete the square, we take half of the coefficient of y , which is 2 − 2 = − 1 , and square it, which is ( − 1 ) 2 = 1 . So, we can rewrite y 2 − 2 y as ( y − 1 ) 2 − 1 .
Substitute back into the original equation Now, we substitute these expressions back into the original equation: ( x + 3 ) 2 − 9 + ( y − 1 ) 2 − 1 + 3 = 0
Simplify the equation We simplify the equation by combining the constants: ( x + 3 ) 2 + ( y − 1 ) 2 − 9 − 1 + 3 = 0 ( x + 3 ) 2 + ( y − 1 ) 2 − 7 = 0 ( x + 3 ) 2 + ( y − 1 ) 2 = 7
Identify the radius Now we have the equation in the standard form ( x + 3 ) 2 + ( y − 1 ) 2 = 7 . Comparing this to ( x − h ) 2 + ( y − k ) 2 = r 2 , we see that r 2 = 7 . Therefore, the radius is r = 7 .
Calculate and round the radius We need to round the radius to the nearest thousandth. We know that 7 ≈ 2.64575 . Rounding to the nearest thousandth, we get 2.646 .
Final Answer The radius of the circle is approximately 2.646 .
Examples
Understanding the radius of a circle is crucial in many real-world applications. For instance, when designing a circular garden, knowing the radius helps determine the amount of fencing needed. If you want a circular garden with an area of 15 m 2 , you can use the area formula A = π r 2 to find the radius r = π A = π 15 ≈ 2.185 m . This ensures you purchase the correct amount of materials and plan the layout accurately.
