Question

question 6 of 10 which of the following circles lie completely within the third quadrant? check all that apply. a. (x + 12)^2+(y + 9)^2 = 9 b. (x + 7)^2+(y + 7)^2 = 4 c. (x + 3)^2+(y + 9)^2 = 82 d. (x + 5)^2+(y + 0)^2 = 7

Explanation:

Step1: Recall circle - equation form

The standard form of a circle is $(x - a)^2+(y - b)^2=r^2$, where $(a,b)$ is the center of the circle and $r$ is the radius. For a circle to be in the third - quadrant, the center $(a,b)$ must have $a\lt0$ and $b\lt0$, and the right - most point $(a + r,b)$, left - most point $(a - r,b)$, top - most point $(a,b + r)$ and bottom - most point $(a,b - r)$ must all have negative $x$ and $y$ coordinates.

Step2: Analyze option A

For the circle $(x + 12)^2+(y + 9)^2=9$, the center is $(-12,-9)$ and the radius $r = 3$. The right - most point is $(-12+3,-9)=(-9,-9)$, the left - most point is $(-12 - 3,-9)=(-15,-9)$, the top - most point is $(-12,-9 + 3)=(-12,-6)$ and the bottom - most point is $(-12,-9 - 3)=(-12,-12)$. All points have negative $x$ and $y$ coordinates, so it lies in the third quadrant.

Step3: Analyze option B

For the circle $(x + 7)^2+(y + 7)^2=4$, the center is $(-7,-7)$ and the radius $r = 2$. The right - most point is $(-7+2,-7)=(-5,-7)$, the left - most point is $(-7 - 2,-7)=(-9,-7)$, the top - most point is $(-7,-7 + 2)=(-7,-5)$ and the bottom - most point is $(-7,-7 - 2)=(-7,-9)$. All points have negative $x$ and $y$ coordinates, so it lies in the third quadrant.

Step4: Analyze option C

For the circle $(x + 3)^2+(y + 9)^2=82$, the center is $(-3,-9)$ and the radius $r=\sqrt{82}\approx9.06$. The right - most point is $(-3 + 9.06,-9)=(6.06,-9)$ which has a positive $x$ - coordinate, so the circle does not lie completely in the third quadrant.

Step5: Analyze option D

For the circle $(x + 5)^2+(y+0)^2=7$, the center is $(-5,0)$ which has a non - negative $y$ - coordinate, so the circle does not lie completely in the third quadrant.

Answer:

A. $(x + 12)^2+(y + 9)^2=9$, B. $(x + 7)^2+(y + 7)^2=4$