Tangent to the parabola `y=x^(2)+ax+1` at the point of intersection of the y-axis also touches the circle `x^(2)+y^(2)=r^(2)`. Also, no point of the parabola is below the x-axis.
The minimum area bounded by the tangent and the coordinate axes is

Options

A. -1
B. 1
C. 0
D. 2

Answer

C

Explanation

(3) The equation of the tangent at (0,1) to the parabola
`y=x^(2)+ax+1` is
y-1=a(x-0)
`orax-y+1=0`
As it touches the circle, we get
`r=(1)/(sqrt(a^(2)+1))`
The radius is maximum when a=0.
Therefore, the equation of the tangent is y=1.
Therefore, the slope of the tangent is 0.