Analyzing an Equation Given in General Form
Consider a circle whose equation $15 x^2+y^2+4 x-6 y-36=0$. 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)
The given equation 15 x 2 + y 2 + 4 x − 6 y − 36 = 0 does not represent a circle because the coefficients of x 2 and y 2 are not equal.
The statement 'To begin converting the equation to standard form, subtract 36 from both sides' is false.
The statement 'To complete the square for the x terms, add 4 to both sides' is false.
All statements about the center and radius are false because the equation is not a circle. None of the statements are true.
Explanation
Analyze the given equation. We are given the equation 15 x 2 + y 2 + 4 x − 6 y − 36 = 0 and asked to determine which statements about the equation are true. Let's analyze the equation to see if it represents a circle. For an equation to represent a circle, the coefficients of the x 2 and y 2 terms must be equal. In this case, the coefficient of x 2 is 15 and the coefficient of y 2 is 1. Since 15 e q 1 , the given equation does not represent a circle. Therefore, any statements about the center and radius of the circle are not applicable.
Analyze the given statements. Now, let's analyze the given statements:
'To begin converting the equation to standard form, subtract 36 from both sides.' To start converting to standard form, we should add 36 to both sides of the equation, not subtract. So, this statement is false.
'To complete the square for the x terms, add 4 to both sides.' To complete the square for the x terms, we first need to isolate the x terms: 15 x 2 + 4 x . Then, we factor out the coefficient of x 2 , which is 15: 15 ( x 2 + 15 4 x ) . To complete the square inside the parenthesis, we need to add ( 2 ⋅ 15 4 ) 2 = ( 15 2 ) 2 = 225 4 . So we add 15 ⋅ 225 4 = 15 4 to both sides of the equation, not 4. Therefore, this statement is also false.
'The center of the circle is at ( − 2 , 3 ) .' Since the equation does not represent a circle, this statement is false.
'The center of the circle is at ( 4 , − 6 ) .' Since the equation does not represent a circle, this statement is false.
'The radius of the circle is 6 units.' Since the equation does not represent a circle, this statement is false.
'The radius of the circle is 49 units.' Since the equation does not represent a circle, this statement is false.
Conclusion. Therefore, none of the given statements are true.
Examples
Understanding the properties of conic sections, like circles, is crucial in various fields. For example, in architecture, knowing the equation of a circle helps in designing circular structures or arches. In physics, understanding circular motion relies on the equation of a circle to describe the path of an object moving in a circular trajectory. In computer graphics, circles are fundamental elements used in creating various shapes and designs. By analyzing the equation of a circle, we can determine its center and radius, which are essential parameters for these applications.
