Question

1. write the standard form of the equation of the circle with the given center and radius. center: (5, - 3), r = 4 reference hw 2.8, #13
2. this is the equation of a circle. x² + y² - 2x - 4y - 31 = 0 reference hw 2.8, #22

a. complete the square and write the given equation in standard form.
b. give the center and radius of the circle. center: __________ radius: r = __________
c. graph the circle.

Explanation:

Step1: Recall circle standard - form

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

Step2: Write equation for first circle

Given center $(5,-3)$ and $r = 4$, substitute $h = 5$, $k=-3$, and $r = 4$ into the standard - form:
$(x - 5)^2+(y+3)^2=16$

Step3: Complete the square for the second circle

For the equation $x^{2}+y^{2}-2x - 4y-31 = 0$.
Group the $x$ and $y$ terms: $(x^{2}-2x)+(y^{2}-4y)=31$.
Complete the square for the $x$ - terms: $x^{2}-2x=(x - 1)^2-1$.
Complete the square for the $y$ - terms: $y^{2}-4y=(y - 2)^2-4$.
Substitute back into the equation: $(x - 1)^2-1+(y - 2)^2-4=31$.
Simplify to get the standard form: $(x - 1)^2+(y - 2)^2=36$.

Step4: Find center and radius of the second circle

From the standard form $(x - 1)^2+(y - 2)^2=36$, the center is $(1,2)$ and the radius $r = 6$.

Step5: Graphing instructions (brief)

To graph $(x - 1)^2+(y - 2)^2=36$, plot the center $(1,2)$ on the coordinate plane. Then, from the center, move 6 units in all directions (up, down, left, right) to plot points on the circle and draw the circle.

Answer:

1. Equation of first circle: $(x - 5)^2+(y + 3)^2=16$
2. a. Standard form of second circle: $(x - 1)^2+(y - 2)^2=36$

b. Center: $(1,2)$; Radius: $r = 6$
c. Plot center $(1,2)$ and points 6 units away to draw the circle.