Quadratic Polynomial Question - Solving for a coefficient using the discriminant

Question

This question has been troubling me:

A parabola whose equation is of the form $y = Bx^2$ (where B is a constant) has the line $20x - y + 20 = 0$ as a tangent. Find $B$.

The explanation says, basically; "The line is a tangent if only one point of contact exists, thus the discriminant $=0$"

I thought that for the discriminant to equal 0, the x-axis had to be tangent to the parabola.

Any help would be appreciated, thanks.

Answer 1

Let us find the intersection $y=Bx^2$ and $20x-y+20=0$

$20x-y+20=0\implies 20x-Bx^2+20=0\implies Bx^2-20x-20=0$ this is a Quadratic Eqaution in $x$

For tangency, both intersection must coincide, so both root of the above equation must be same, i.e., we need the discriminant to be $0$

So, we need $(-20)^2-4\cdot B\cdot(-20)=0$