91Ó°ÊÓ

Given that the coordinates are 5 units from (-8, 1).

We have to find the twelve points with integer coordinates.

$(−8,−6),(−8,−4),(−11,5),(−11,−3),(−12,4),(−12,−2),(−3,1),(−13,1),(−4,4),(−4,−2),(−5,5),(−5,−3)$The given point is (-8, 1),

We will add suitable points in order to get the points that are 5 units from (-8, 1).

First point:

$\begin{array}{l}{C}_1=\left(−8,1\right)+\left(0,5\right)\\ C_1=\left(−8,−6\right)\end{array}$

Second point:

$\begin{array}{l}{C}_2=\left(−8,1\right)+\left(0,−5\right)\\ C_2=\left(−8,−4\right)\end{array}$

Third point:

$\begin{array}{l}{C}_3=\left(−8,1\right)+\left(−3,4\right)\\ C_3=\left(−11,5\right)\end{array}$

Fourth point:

$\begin{array}{l}{C}_4=\left(−8,1\right)+\left(−3,−4\right)\\ C_4=\left(−11,−3\right)\end{array}$

Fifth point:

$\begin{array}{l}{C}_5=\left(−8,1\right)+\left(−4,3\right)\\ C_5=\left(−12,4\right)\end{array}$

Sixth point:

$\begin{array}{l}{C}_6=\left(−8,1\right)+\left(−4,−3\right)\\ C_6=\left(−12,−2\right)\end{array}$

Seventh point:

$\begin{array}{l}{C}_7=\left(−8,1\right)+\left(5,0\right)\\ C_7=\left(−3,1\right)\end{array}$

Eighth point:

$\begin{array}{l}{C}_8=\left(−8,1\right)+\left(−5,0\right)\\ C_8=\left(−13,1\right)\end{array}$

Nineth point:

$\begin{array}{l}{C}_9=\left(−8,1\right)+\left(4,3\right)\\ C_9=\left(−4,4\right)\end{array}$

Tenth point:

$\begin{array}{l}{C}_{10}=\left(−8,1\right)+\left(4,−3\right)\\ C_{10}=\left(−4,−2\right)\end{array}$

Eleventh point:

$\begin{array}{l}{C}_{11}=\left(−8,1\right)+\left(3,4\right)\\ C_{11}=\left(−5,5\right)\end{array}$

Twelfths point.

$\begin{array}{l}{C}_{12}=\left(−8,1\right)+\left(3,−4\right)\\ C_{12}=\left(−5,−3\right)\end{array}$

So, the twelve points from the point (-8, 1) are:

Given that the coordinates are 5 units from (-8, 1).

We have to write an equation of the circle containing these points.

Here, center = (h, k) = (-8, 1) and the radius = r = 5.

Plug the values in the standard form of a circle:

${\left(x−h\right)}^2+{\left(y−k\right)}^2=r^2$

$\begin{array}{l}{(x+8)}^2+{\left(y−1\right)}^2=5^2\\ {(x+8)}^2+{\left(y−1\right)}^2=25\end{array}$

So, the equation is${(x+8)}^2+{\left(y−1\right)}^2=25$