Respuesta :
[tex](x-7)^{2} + (y+6)^{2} = 10[/tex] is the equation of the required circle.
Step-by-step explanation:
Step 1 :
Let A = (8,-9) and B = (6,-3) be the endpoints of the diameter of the given circle.
We need to determine the circle's equation with this diameter
Step 2 :
The mid point of the diameter will be center of the required circle.
Let M = (x,y) be the mid point of the circle.
So we have x = (8+6)÷ 2 , y = ((-9) + (-3)) ÷ 2
M = ( 7, -6) is the midpoint of the diameter and the center of the circle.
Step 3 :
The radius of the circle is distance between midpoint and any one of the end point of the diameter that is distance between point A(8,-9) and M ( 7,-6)
Distance between the 2 points [tex](x_{1} ,y_{1}) and (x_{2} ,y_{2})[/tex] is given by
[tex]\sqrt{(x_{2}-x_{1})^{2}+ (y_{2}-y_{1})^{2} }[/tex]
So distance between A(8,-9) and M ( 7,-6) = [tex]\sqrt{(7-8)^{2}+ (-6-(-9))^{2} }[/tex]
= [tex]\sqrt{(1)^{2}+ (3)^{2} } = \sqrt{10 }[/tex]
Hence the radius r is [tex]\sqrt{10}[/tex]
Step 4 :
The equation of circle with center (a,b) = (7,-6) and radius r = [tex]\sqrt{10}[/tex] is given by
[tex](x-a)^{2} + (y-b)^{2} = r^{2}[/tex]
=> [tex](x-7)^{2} + (y-(-6))^{2} = (\sqrt{10} )^{2}[/tex]
=> [tex](x-7)^{2} + (y+6)^{2} = 10[/tex] is equation of the required circle.
Step 5 :
Answer :
[tex](x-7)^{2} + (y+6)^{2} = 10[/tex] is equation of the required circle.
