Question

Write the equation of the circle given the following information

Answer 1

Given;

There are given that points are:

`(1,13)\text{ and \lparen-3,-9\rparen}`

Step-by-step explanation:

From the standard form of the circle:

`(x-a)^2+(y-b)^2=r^2`

Where

a and b represent the center.

Now,

To establish the equation, we require to know it is center and radius.

Since we are given the endpoints of the diameter

Then,

The center will be at the midpoint and the radius will be the distance from the center to either of the two given points.

Then,

From the formula of midpoint to calculate the midpoint:

`(x,y)=((x_1+x_2)/(2),(y_1+y_2)/(2))`

Then,

From the given two points:

`\begin{gathered} (x,y)=((1-3)/(2),(13-9)/(2)) \\ (x,y)=(-(2)/(2),(4)/(2)) \\ (x,y)=(-1,2) \end{gathered}`

The midpoint is ( -1, 2).

Now,

We need to find the value of the radius.

So,

To calculate the radius, we need to use the distance formula:

Then,

From the formula of distance, here we will use the points: (-1, 2) and (1, 13)

`\begin{gathered} r=√((x_2-x_1)^2+(y_2-y_1)^2) \\ r=√(\left(1+1\right)^2+\left(13-2\right)^2) \\ r=√(4+121) \\ r=√(125) \end{gathered}`

Now,

We have the value of radius and the point of center.

Then,

Put the value of radius and point of the center into the standard form of the circle:

So,

From the standard form of the circle:

`\begin{gathered} (x-a)^(2)+(y-b)^(2)=r^(2) \\ (x-\left(-1\right))^2+(y-2)^2=\lparen√(125)^)^2 \\ (x+1)^2+(y-2)^2=125 \end{gathered}`

Final answer:

Hence, the equation of the circle is shown below;

`(x+1)^(2)+(y-2)^(2)=125`