Consider a circle whose equation is $x^2+y^2+4 x-6 y-36=0$. Which statements are true? Check all that apply.
A. To begin converting the equation to standard form, subtract 36 from both sides.
B. To complete the square for the $x$ terms, add 4 to both sides.
C. The center of the circle is at $(-2,3)$.
D. The center of the circle is at $(4,-6)$.
E. The radius of the circle is 6 units.
F. The radius of the circle is 49 units.
Answer (1)
Convert the given circle equation to standard form by completing the square for both x and y terms.
Identify the center ( h , k ) and radius r from the standard form equation ( x − h ) 2 + ( y − k ) 2 = r 2 .
Evaluate each statement based on the derived center and radius.
Determine the true statements: adding 4 to both sides completes the square for x terms, and the center is at ( − 2 , 3 ) .
True statements: To complete the square for the x terms, add 4 to both sides. The center of the circle is at ( − 2 , 3 ) .
Explanation
Analyze the problem and given data We are given the equation of a circle: x 2 + y 2 + 4 x − 6 y − 36 = 0 . Our goal is to determine which of the provided statements about this circle are true. To do this, we will convert the given equation into the standard form of a circle's equation, which is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) represents the center of the circle and r is its radius.
Evaluate the first statement The first statement says: 'To begin converting the equation to standard form, subtract 36 from both sides.' Let's analyze this. The original equation is x 2 + y 2 + 4 x − 6 y − 36 = 0 . To start completing the square, we want to isolate the constant term on the right side. So, we should add 36 to both sides, not subtract. Therefore, this statement is false.
Evaluate the second statement The second statement says: 'To complete the square for the x terms, add 4 to both sides.' Let's verify this. We have the terms x 2 + 4 x . To complete the square, we take half of the coefficient of x , which is 4/2 = 2 , and square it: 2 2 = 4 . So, we need to add 4 to complete the square for the x terms. Thus, the statement is true.
Convert the equation to standard form Now, let's complete the square for both x and y terms in the equation x 2 + y 2 + 4 x − 6 y − 36 = 0 .
First, add 36 to both sides: x 2 + y 2 + 4 x − 6 y = 36 .
Next, complete the square for the x terms by adding ( 4/2 ) 2 = 4 to both sides: x 2 + 4 x + 4 + y 2 − 6 y = 36 + 4 .
Then, complete the square for the y terms by adding ( − 6/2 ) 2 = 9 to both sides: x 2 + 4 x + 4 + y 2 − 6 y + 9 = 36 + 4 + 9 .
Rewrite the equation in standard form: ( x + 2 ) 2 + ( y − 3 ) 2 = 49 .
Identify the center and radius From the standard form ( x + 2 ) 2 + ( y − 3 ) 2 = 49 , we can identify the center and radius of the circle.
The center of the circle is ( h , k ) = ( − 2 , 3 ) .
The radius of the circle is r = 49 = 7 .
Evaluate the remaining statements The third statement says: 'The center of the circle is at ( − 2 , 3 ) '. This matches our calculated center, so this statement is true.
The fourth statement says: 'The center of the circle is at ( 4 , − 6 ) '. This does not match our calculated center, so this statement is false.
The fifth statement says: 'The radius of the circle is 6 units.' This does not match our calculated radius, which is 7, so this statement is false.
The sixth statement says: 'The radius of the circle is 49 units.' This is the value of r 2 , not r , so this statement is false.
Final Answer Therefore, the true statements are:
To complete the square for the x terms, add 4 to both sides.
The center of the circle is at ( − 2 , 3 ) .
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, consider designing a circular garden with a specific radius and center point. By knowing the equation of the circle, you can accurately map out the garden's boundaries on a coordinate plane, ensuring it fits perfectly within your yard. This also helps in calculating the amount of fencing needed or the area to be covered with plants, making your gardening project precise and efficient.
