The centre and radius of the circle x2 + y2 + 2gx + 2yf + c=0 are
The general equation of a circle is given x2 + y2 + 2gx + 2yf + c=0 . The centre and radius of this circle are (−g,−f) and √g2 + f2 − c respectively ; We will try to get this result.
We know the equation of a circle whose centre is (h,k) and radius r, is (x−h)2 + (y−k)2 = r2
We will simplify this and compare it with x2 + y2 + 2gx + 2yf + c=0 to find radius and centre.
⇒ x2 + y2 − 2hx − 2ky + h2 + k2 − r2 = 0 - - - - - (1)
x2 + y2 + 2gx + 2yf + c=0 - - - - - (2)
(1) and (2) represents the same circle.
⇒ -h = g and -k = f
⇒ (h,k) ≡ (−g,−f)is the centre.We also have c = h2 + k2 − r2 = 0
(h,k)=(−g,−f)
⇒ c = (−g)2 + (−f)2 − r2
⇒ r = √g2 + f2 − c