The radius of the circle S is same as the radius of x2 + y2 − 2x + 4y − 11 = 0 and the centre of S is the centre of x2 + y2 − 2x − 4y + 11 = 0 . Find the equation of S.
To find the equation of a circle, we need the centre and radius of the circle, In this problem, we are given two circles, using which we can find the centre and radius.
To find the radius
x2 + y2 − 2x + 4y − 11 = 0 and S(our circle) has the same radius.
We know the radius of the circle x2 + y2 + 2gx + 2fy + c=0 is
√g2 + f2 − c
⇒ r = √12 + 22 + 11
=4
To find the centre:
We are given S and x2 + y2 − 2x − 4y + 11 = 0 have the same centre
Centre of x2 + y2 + 2gx + 2fy + c=0 is (−g,−f).
⇒ centre of x2 + y2 − 2x − 4y + 11 = 0 is (1,2)
We got the centre (1,2) and radius (4) of the circle.
⇒ Equation is (x−1)2 + (y−2)2 = 42
⇒ x2 + y2 − 2x − 4y + 1 + 4 = 16
⇒ x2 + y2 − 2x − 4y − 11 = 0