Respuesta :
The equation of a circle is given by the expression
[tex](x-h)^2+(y-k)^2=r^2[/tex]Where (h,k) is the center and r is the radius.
We have 2 points that happen to define the diameter of the circle. In order to obtain the radius we need the distance between those 2 points:
[tex]D=d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]This is the formula to obtain the distance between two points on the plane. We proceed by filling that formula with the information given by the problem
[tex]D=\sqrt[]{(6-4)^2+(-5-3)^2}=\sqrt[]{2^2+(-8)^2}=\sqrt[]{68}=\sqrt[]{4\cdot17}=2\sqrt[]{17}[/tex]That's the diameter, the radius is D/2
[tex]r=\frac{D}{2}=\frac{2\sqrt[]{17}}{2}=\sqrt[]{17}[/tex]Finally, we need to find the center of the circumference. For this, we need the middle point of the segment that joins the two given points since we know those define a diameter.
The middle point is given by the formula
[tex]M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]Using the information of our particular problem, we get
[tex]\text{Center}=M=(\frac{4+6}{2},\frac{3-5}{2})=(\frac{10}{2},\frac{-2}{2})=(5,-1)[/tex]So, the center of the circle is (5,-1)
Finally, we have everything we need to use the first formula in the explanation:
(h,k)=(5,-1) and r=(17)^(1/2)
[tex](x-5)^2+(y+1)^2=17[/tex]This last result is the answer to our problem
