Question

Given: Circle O with diameter CDC (-7, -1) and D (1,2)Create the equation of this circle,+-:: (2+3):: (y – 1):: (y + 1):: 25:: 100

Answer 1

`(x+3)^2+(y+1)^2=25`

Step-by-step explanation

The equation of a circle with center (h,k) and radius r units is given by:

`(x-h)^2+(y-k)^2=r^2`

then

Step 1

find the diameter of the cirlce:

to do this, we can use the distance between two points formula:

if

`\begin{gathered} A\mleft(x_1,y_1\mright) \\ B(x_2,y_2) \end{gathered}`

the distance from A to B is

`\begin{gathered} d_(AB)=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ \end{gathered}`

so,let

distance CD=

`\begin{gathered} CD=\sqrt[]{(1-(-7))^2+(2-(-4))^2} \\ CD=\sqrt[]{(8)^2+(6)^2} \\ CD=\sqrt[]{64+36} \\ CD=\sqrt[]{100} \\ CD=10 \end{gathered}`

hence, the diameter of the circle is

`\begin{gathered} \text{diameter}=10 \\ w\text{e know also} \\ \text{diameter}=2\cdot\text{raidus} \\ \frac{\text{diamteter}}{2}=radius \\ \text{replace} \\ (10)/(2)=\text{ radius} \\ \text{radius}=5 \end{gathered}`

Step 2

find the center of the circle:

the center of the circle is the midpoint of CD

so

`\text{midpoint}=((x_1+x_2)/(2),(y_1+y_2)/(2))`

replace

`\begin{gathered} \text{midpoint}=((x_1+x_2)/(2),(y_1+y_2)/(2)) \\ \text{midpoint}=((-7+1)/(2),\frac{-4+2_{}}{2}) \\ \text{midpoint}=((-6)/(2),(-2)/(2)) \\ \text{midpoint}=(-3,-1) \\ \end{gathered}`

so, the center of the circle is (-3,-1)

Step 3

finally, replace in the formula to get the equation of the circle

let

`\begin{gathered} center=\mleft(-3,-1\mright) \\ radius=5 \end{gathered}`

replace

`\begin{gathered} (x-h)^2+(y-k)^2=r^2 \\ (x-(-3))^2+(y-(-1))^2=5^2 \\ (x+3)^2+(y+1)^2=25 \end{gathered}`

therefore, the answer is

`(x+3)^2+(y+1)^2=25`

I hope this helps you