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 (2)
Rewrite the given equation by grouping the x and y terms together: ( x 2 + 4 x ) + ( y 2 − 6 y ) = 36 .
Complete the square for the x terms: ( x + 2 ) 2 + ( y 2 − 6 y ) = 40 .
Complete the square for the y terms: ( x + 2 ) 2 + ( y − 3 ) 2 = 49 .
Identify the center of the circle as ( − 2 , 3 ) and the radius as 7 . The true statements are that to complete the square for the x terms, add 4 to both sides and the center of the circle is at ( − 2 , 3 ) . T r u e , T r u e
Explanation
Analyze the problem and rewrite the equation in standard form 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 given statements about this circle are true. To do this, we will rewrite the equation in standard form, which is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center of the circle and r is the radius.
Group x and y terms First, let's rewrite the given equation by grouping the x and y terms together: ( x 2 + 4 x ) + ( y 2 − 6 y ) = 36
Complete the square for x terms Now, we'll complete the square for the x terms. To do this, we take half of the coefficient of the x term (which is 4), square it (which gives us 2 2 = 4 ), and add it to both sides of the equation: ( x 2 + 4 x + 4 ) + ( y 2 − 6 y ) = 36 + 4 ( x + 2 ) 2 + ( y 2 − 6 y ) = 40
Complete the square for y terms Next, we'll complete the square for the y terms. We take half of the coefficient of the y term (which is -6), square it (which gives us ( − 3 ) 2 = 9 ), and add it to both sides of the equation: ( x + 2 ) 2 + ( y 2 − 6 y + 9 ) = 40 + 9 ( x + 2 ) 2 + ( y − 3 ) 2 = 49
Identify the center and radius Now, the equation is in standard form: ( x + 2 ) 2 + ( y − 3 ) 2 = 49 . From this, we can identify the center of the circle as ( − 2 , 3 ) and the radius as 49 = 7 .
Evaluate each statement Now, let's evaluate each statement:
Statement 1: To begin converting the equation to standard form, subtract 36 from both sides. This is incorrect. We should add 36 to both sides to isolate the constant term. Thus, this statement is False .
Statement 2: To complete the square for the x terms, add 4 to both sides. This is correct, as we added 4 to both sides to complete the square for the x terms. Thus, this statement is True .
Statement 3: The center of the circle is at ( − 2 , 3 ) .
This is correct based on our standard form equation. Thus, this statement is True .
Statement 4: The center of the circle is at ( 4 , − 6 ) .
This is incorrect. Thus, this statement is False .
Statement 5: The radius of the circle is 6 units. This is incorrect; the radius is 7 units. Thus, this statement is False .
Statement 6: The radius of the circle is 49 units. This is incorrect; 49 is the square of the radius, not the radius itself. Thus, 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, civil engineers use circle equations to design tunnels and bridges, ensuring structural integrity and precise alignment. Architects apply these principles to create aesthetically pleasing and structurally sound circular domes and arches. In navigation, the equation of a circle helps determine the range of a radar system or the coverage area of a cell tower, ensuring efficient and reliable service.
The true statements are that to complete the square for the x terms, we add 4 to both sides (Statement B), and that the center of the circle is at ( − 2 , 3 ) (Statement C). The other statements regarding the radius and center location are false.
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