Answer :
Answer: [tex](x+8)^{2}[/tex] + [tex](y+3)^{2}[/tex] = [tex]3^{2}[/tex]
Step-by-step explanation:
The equation of a circle is given as :
[tex](x-a)^{2}[/tex] + [tex](y-b)^{2}[/tex] = [tex]r^{2}[/tex]
Where (a,b) are the coordinate of the center and r is the radius of the circle.
The end point of the diameter is give as (-5, -3 ) and ( -11 , -3 ) , This means that we can find the coordinate of the center by finding the mid point of the end point. The mid point is calculated by :
Mid point = ([tex]\frac{x_{1}+x_{2}}{2}[/tex] , [tex]\frac{y_{1}+y_{2}}{2}[/tex] )
[tex]x_{1}[/tex] = -5
[tex]x_{2}[/tex] = -11
[tex]y_{1}[/tex] = -3
[tex]y_{2}[/tex] = -3
Substituting this values into the formula for finding mid point , we have
Mid point = ([tex]\frac{-5 - 11}{2}[/tex] , [tex]\frac{-3 - 3}{2}[/tex]
Mid-point = (-8 , -3)
Remember that Radius is half of the diameter , To find the diameter we must find the distance between the two end point using the formula for calculating distance between two points , that is
D = [tex]\sqrt{(x_{2}-x_{1}) ^{2}+(y_{2}-y_{1}) ^{2}}[/tex]
Substituting the values :
D = [tex]\sqrt{(-11+5)^{2}+(-3+3)^{2}}[/tex]
D = [tex]\sqrt{36}[/tex]
D = 6
Therefore , The diameter i s 6
And since radius is half of the diameter , radius is thus
r = 6/2
r = 3
So , substituting the values gotten into the equation of circle , we have:
[tex](x-a)^{2}[/tex] + [tex](y-b)^{2}[/tex] = [tex]r^{2}[/tex]
[tex](x+8)^{2}[/tex] + [tex](y+3)^{2}[/tex] = [tex]3^{2}[/tex]