Which of the following is a property of the equation of a circle?
A. There is a term in [tex]$x y$[/tex].
B. The degree of the expression in [tex]$x$[/tex] and [tex]$y$[/tex] is 2
C. The coefficients of [tex]$x$[/tex] and [tex]$y$[/tex] are the same.
D. The center is always at the origin.
Answer (2)
The properties of the equation of a circle indicate that the coefficients of x 2 and y 2 are the same, which denotes the uniform scaling of the circle. Thus, the correct answer is option C . Other options either describe qualities that are not universally true for circles or are true for multiple conic sections.
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The equation of a circle ( x − a ) 2 + ( y − b ) 2 = r 2 expands to x 2 + y 2 − 2 a x − 2 b y + ( a 2 + b 2 − r 2 ) = 0 .
The equation does not have an x y term.
The degree of the expression is 2.
The coefficients of x 2 and y 2 are the same.
The center is not always at the origin. Thus, the correct property is that the coefficients of x 2 and y 2 are the same: C .
Explanation
Analyze the general equation of a circle Let's analyze the properties of the equation of a circle to determine the correct answer. The general equation of a circle with center ( a , b ) and radius r is given by: ( x − a ) 2 + ( y − b ) 2 = r 2
Expand the equation Expanding this equation, we get: x 2 − 2 a x + a 2 + y 2 − 2 b y + b 2 = r 2 Rearranging the terms, we have: x 2 + y 2 − 2 a x − 2 b y + ( a 2 + b 2 − r 2 ) = 0
Examine each option Now, let's examine each option:
A. There is a term in x y . The expanded equation does not contain an x y term. So, this is not a general property of the equation of a circle.
B. The degree of the expression in x and y is 2. The highest power of both x and y in the equation is 2. This seems to be a property of the equation of a circle.
C. The coefficients of x and y are the same. In the expanded equation, the coefficients of x 2 and y 2 are both 1 (and therefore the same). This also appears to be a property of the equation of a circle.
D. The center is always at the origin. The center of the circle is at ( a , b ) , which is only at the origin if a = 0 and b = 0 . This is not always the case.
Determine the correct property Comparing options B and C, option C is a more precise property. Option B is true, but it's also true for other conic sections like ellipses and hyperbolas. Option C specifically describes a circle.
Therefore, the correct answer is that the coefficients of x 2 and y 2 are the same.
Examples
Understanding the properties of a circle's equation is crucial in various fields. For example, in computer graphics, knowing that the coefficients of x 2 and y 2 are the same allows developers to efficiently render circles on the screen. Similarly, in physics, when analyzing circular motion, the equation of the circular path relies on these properties to accurately model the movement of objects.
