Let P(2t,t2) be any point on the parabola. Centre of the given circle C=(−g,−f)=(−3,0) For PC to be minimum, it must be the normal to the parabola at P. Slope of line PC=
y2−y1
x2−x1
=
t2−0
2t+3
Also, slope of tangent to parabola at P=
dy
dx
=
x
2
=t ∴ Slope of normal =
−1
t
∴
t2−0
2t+3
=
−1
t
⇒t3+2t+3=0 ⇒(t+1)(t2−t+3)=0 ∴ Real roots of above equation is t=−1 Coordinate of P=(2t,t2)=(−2,1) Slope of tangent to parabola at P=t=−1 Therefore, equation of tangent is: (y−1)=(−1)(x+2) ⇒x+y+1=0