Which of the following is a degenerate circle?
A. $x^2+y^2=-2$
B. $x+y=7$
C. $x^2+y^2=5$
D. $(x-5)^2+(y-3)^2=0$
Answer (1)
A degenerate circle has a radius of 0.
Option A has a negative radius squared, which is not possible.
Option B is a linear equation, not a circle.
Option C has a non-zero radius.
Option D has a radius of 0, thus it's a degenerate circle: ( x − 5 ) 2 + ( y − 3 ) 2 = 0 .
Explanation
Understanding Degenerate Circles A degenerate circle is a circle with a radius of 0. The general equation of a circle is ( x − a ) 2 + ( y − b ) 2 = r 2 , where ( a , b ) is the center and r is the radius. We need to check each option to see if it represents a circle with radius 0.
Analyzing Option A Option A: x 2 + y 2 = − 2 . This can be written as ( x − 0 ) 2 + ( y − 0 ) 2 = − 2 . Since r 2 = − 2 , the radius would be an imaginary number, which is not possible for a real circle. So, this is not a valid circle.
Analyzing Option B Option B: x + y = 7 is the equation of a line, not a circle.
Analyzing Option C Option C: x 2 + y 2 = 5 . This can be written as ( x − 0 ) 2 + ( y − 0 ) 2 = 5 . Here, r 2 = 5 , so r = 5 . This is a circle with a non-zero radius.
Analyzing Option D Option D: ( x − 5 ) 2 + ( y − 3 ) 2 = 0 . This can be written as ( x − 5 ) 2 + ( y − 3 ) 2 = 0 2 . Here, r 2 = 0 , so r = 0 . This represents a degenerate circle centered at ( 5 , 3 ) .
Conclusion Therefore, the degenerate circle is option D.
Examples
Imagine you're designing a dartboard. A standard dartboard has circular regions, but a degenerate circle is like a single point on the board. It's a circle that has shrunk to have a radius of zero. Understanding degenerate circles helps in recognizing special cases in geometry and algebra, and it reinforces the concept of a circle's equation and its parameters.
