Hence, equation of circle is:
[tex]x^2+y^2-4y=48[/tex]
Te end points of the diameter are given to be:
(−6, 6) and (6, −2).
We know that the center of the circle is the mid-point of the diameter.
and the coordinates of the mid-point (e,f) of (a,b) and (c,d) is calculated as:
[tex]e=\dfrac{a+c}{2}, f=\dfrac{b+d}{2}[/tex]
Here we have (a,b)=(-6,6) and (c,d)=(6,-2).
Hence, the coordinates of center (e,f) is calculated as:
[tex]e=\dfrac{-6+6}{2},f=\dfrac{6-2}{2}\\\\e=\dfrac{0}{2},f=\dfrac{4}{2}\\\\e=0,f=2[/tex]
Hence, center is located at: (0,2)
Now, the radius of circle is the the distance between the center and a point on the circle.
i.e. distance between (0,2) and (6,-2).
We know that the distance between two points (a,b) and (c,d) is calculated as:
[tex]=\sqrt{(c-a)^2+(d-b)^2}[/tex]
Here we have (a,b)=(0,2) and (c,d)=(6,-2)
Hence, the length of the radius is calculated as:
[tex]\sqrt{(6-0)^2+(-2-2)^2}\\\\=\sqrt{36+16}\\\\=\sqrt{52}\\\\=2\sqrt{13}[/tex]
Hence, the equation of the circle with center (h,k) and radius r is given by:
[tex](x-h)^2+(y-k)^2=r^2[/tex]
Hence, the equation of circle is:
[tex](x-0)^2+(y-2)^2=(2\sqrt{13})^2\\\\x^2+y^2+4-4y=52\\\\x^2+y^2-4y=48[/tex]
Hence, equation of circle is:
[tex]x^2+y^2-4y=48[/tex]
