Let P be a point on the parabola x²=4y. If the distance of P from the centre of the circle x²+y²+6x+8=0 is minimum, then the equation of the tangent to the parabola at P, is

Let P(2t,t²) be any point on the parabola. Centre of circle = (-g,-f) = (-3,0). For the distance between point P and center of the circle to be minimum, the line is drawn from the center of the circle to point P must be normal to the parabola at P. Slope of line joining centre of circle to point P = (t²-0)/(2t+3). Slope of tangent to parabola at P = dy/dx = x/2 = t. Slope of normal = -1/t. Therefore, (t²)/(2t+3) = -1/t => t³+2t+3=0 => (t+1)(t²-t+3)=0. Real roots of the above equation are t=-1. Coordinate of P = (2t,t²) = (-2,1). Slope of tangent to parabola at P = dy/dx = x/2 = t = -1. Therefore, the equation of tangent is (y-1) = (-1)(x+2) => x+y+1=0. Hence, the answer is option (C).