The general form of the equation of a circle is x2+y2−4x−8y−5=0.
What are the coordinates of the center of the circle?
A circle can be defined as the locus of all points that satisfy the equation
(x−h)2+(y−k)2=r2 ( Standard Form )
where r is the radius of the circle,
and h,k are the coordinates of its center.
The general Form is:
x2+y2+ax+by+c=0
Standard Form to general Form:
(x−h)2+(y−k)2=r2x2−2xh+h2+y2−2yk+k2=r2x2+y2+x⋅(−2h)⏟=a+y⋅(−2k)⏟=b+h2+k2−r2⏟=c=0
h,k and r ?
x2+y2+x⋅(−2h)⏟=a+y⋅(−2k)⏟=b+h2+k2−r2⏟=c=0a=−2hh=−a2b=−2kk=−b2c=h2+k2−r2c=(−a2)2+(−b2)2−r2c=a2+b24−r2r2=a2+b24−cr=√a2+b24−c
If we have a,b and c, we can calculate h,k and r:
x2+y2+ax+by+c=0h=−a2k=−b2r=√a2+b24−c
a=−4b=−8c=−5
x2+y2−4x−8y−5=0h=−−42h=2k=−−82k=4r=√(−4)2+(−8)24−(−5)r=√16+644+5r=√20+5r=√25r=5
The coordinates of the center of the circle is (2,4) and the radius is 5
