Consider a circle whose equation is [tex]$x^2+y^2+4 x-6 y-36=0$[/tex]. Which statements are true? Check all that apply.
A. To begin converting the equation to standard form, subtract 38 from both sides.
B. To complete the square for the [tex]$x$[/tex] terms, add 4 to both sides.
C. The center of the circle is at ( [tex]$-2,3$[/tex] ).
D. The center of the circle is at ([tex]$4,-6$[/tex]).
E. The radius of the circle is 6 units.
F. The radius of the circle is 49 units.
Answer (1)
Convert the general form of the circle's 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 .
Determine the radius by taking the square root of the constant term on the right side of the standard equation.
Conclude that the correct statements are: adding 4 to both sides completes the square for x , and the center is at ( − 2 , 3 ) .
Add 4 to both sides, Center is at ( − 2 , 3 )
Explanation
Analyze the problem and convert to standard form We are given the equation of a circle in general form: x 2 + y 2 + 4 x − 6 y − 36 = 0 . Our goal is to convert this equation to 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 To convert the given equation to standard form, we need to complete the square for both the x and y terms. Let's start 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 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 is 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 complete the square for the y terms. We take half of the coefficient of the y term (which is -6), square it (which is ( − 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 center and radius Now the equation is in standard form: ( x + 2 ) 2 + ( y − 3 ) 2 = 49 . From this, 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 statements Now, let's evaluate the given statements:
To begin converting the equation to standard form, subtract 38 from both sides: This is incorrect. We added values to both sides to complete the square.
To complete the square for the x terms, add 4 to both sides: This is correct, as shown in step 3.
The center of the circle is at ( − 2 , 3 ) : This is correct, as identified in step 5.
The center of the circle is at ( 4 , − 6 ) : This is incorrect.
The radius of the circle is 6 units: This is incorrect. The radius is 7 units.
The radius of the circle is 49 units: This is incorrect. 49 is the square of the radius.
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, when designing a circular garden, knowing the center and radius helps in accurately planning the layout and determining the amount of fencing needed. Similarly, in architecture, circular arches and domes rely on the principles of circle geometry for structural integrity and aesthetic appeal. The ability to convert between general and standard forms of a circle's equation allows engineers and designers to easily work with circular shapes in diverse projects, from creating efficient irrigation systems to designing visually stunning buildings.
