Question

question 1 of 10
which of the following is a degenerate circle?
a. $x + y=2$
b. $x^{2}+y^{2}=-1$
c. $(x - 4)^{2}+(y - 2)^{2}=0$
d. $x^{2}+y^{2}=1$

Explanation:

Step1: Recall circle equation

The standard form of a circle equation is $(x - a)^2+(y - b)^2=r^2$, where $(a,b)$ is the center and $r$ is the radius.

Step2: Analyze option A

$x + y=2$ is a linear equation, not a circle equation.

Step3: Analyze option B

For $x^{2}+y^{2}=- 1$, since $x^{2}\geq0$ and $y^{2}\geq0$ for real - valued $x$ and $y$, the left - hand side is non - negative, and this equation has no real solutions. It does not represent a circle.

Step4: Analyze option C

For $(x - 4)^2+(y - 2)^2=0$, the sum of two non - negative real numbers $(x - 4)^2$ and $(y - 2)^2$ is zero if and only if $x-4 = 0$ and $y - 2=0$, i.e., $x = 4$ and $y = 2$. This represents a single point $(4,2)$, which is a degenerate circle (a circle with radius $r = 0$).

Step5: Analyze option D

$x^{2}+y^{2}=1$ is a circle with center at the origin $(0,0)$ and radius $r = 1$.

Answer:

C. $(x - 4)^2+(y - 2)^2=0$