Question

question 4

1. if a circle is centered at (5, -7) and goes through the point (-3, -1), what is the radius?

15 pts
question 5

1. given the equation of a circle, identify the center and the radius.

(x - 9)^2+(y + 4)^2 = 36
center:
radius:

Explanation:

Step1: Recall circle - equation formula

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

Step2: Identify center for Question 5

For the equation $(x - 9)^2+(y + 4)^2=36$, comparing with the standard - form, we have $h = 9$ and $k=-4$. So the center is $(9,-4)$.

Step3: Identify radius for Question 5

Since $(x - 9)^2+(y + 4)^2=36$ and $r^2 = 36$, then $r=\sqrt{36}=6$.

Step4: Recall distance formula for Question 4

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. The radius of a circle is the distance between the center $(x_1,y_1)=(5,-7)$ and the point on the circle $(x_2,y_2)=(-3,-1)$.

Step5: Calculate radius for Question 4

Substitute $x_1 = 5,y_1=-7,x_2=-3,y_2=-1$ into the distance formula:
\[

$$\begin{align*} r&=\sqrt{(-3 - 5)^2+(-1+7)^2}\\ &=\sqrt{(-8)^2+6^2}\\ &=\sqrt{64 + 36}\\ &=\sqrt{100}\\ &=10 \end{align*}$$

\]

Answer:

Question 5 - Center: $(9,-4)$, Radius: $6$
Question 4 - Radius: $10$