Question

$y=f\left(x\right)$ is the parabola of the form $y=x^2+ax+1,$ its tangent at the point of intersection of $y-$axis and parabola also touches the circle $x^2+y^2=r^2.$ It is known that no point of the parabola is below $x-$axis. The radius of the circle (in units) when $a$ attains its maximum value is

• A. $\frac{1}{\sqrt{10}}$
• B. $\frac{1}{\sqrt{5}}$
• C. $1$
• D. $\sqrt{5}$

Answer 1

The correct option is B $\frac{1}{\sqrt{5}}$

Since no point of the parabola is below $x-$axis.
$\therefore a^2-4\le 0$
$\therefore$ maximum value of $a$ is $2.$
Equation of parabola when $a=2$ is $y=x^2+2x+1$
It intersects $y-$axis at $(0,1).$
Equation of the tangent at $(0,1)$ is $y=2x+1$
Since, $y=2x+1$ touches the circle $x^2+y^2=r^2$
$r=\frac{1}{\sqrt{a^2+1}}$
$\therefore r=\frac{1}{\sqrt{5}}$units