Question

Let $y=f\left(x\right)$ be a parabola of the form $y=x^2+ax+1$ such that no point of the parabola is below $x-$axis. If its tangent at the point of intersection with $y-$axis also touches the circle $x^2+y^2=r^2$, then the slope of the tangent when radius of the circle is maximum is

Answer 1

Equation of parabola is $y=x^2+ax+1$
It intersects $y-$axis at $(0,1).$
Equation of the tangent at $(0,1)$ to the parabola $y=x^2+ax+1$ is $\frac{y+1}{2}=\frac{a}{2}(x+0)+1$
i.e. $ax-y+1=0$
$\therefore r=\frac{1}{\sqrt{a^2+1}}$
radius is maximum when $a=0$
$\therefore$ Equation of tangent is $y=1$
So, slope of tangent is $0$