Algebra, points of intersection

Why does 'a' have 2 values if y= x is tangent to parabola y = x^2 + ax + 1. I thought there would be only 1 solution.

Similarly, why are there 2 valus of 'm' when y = mx - 0.5 is tangent to y = x^2 + 5x?

There is only 1 solution for b when y = -x is tangent to y = x^2 + x + b ??

Re: Algebra, points of intersection

Please explain how did you proceed?

Re: Algebra, points of intersection

I used the discriminant to find the unknown. I equated the discriminant to 0 as the line is a tangent to the parabola.

3 Attachment(s)

Re: Algebra, points of intersection

In order that a line, y= mx+ n, be tangent to a parabola, $y= ax^2+ bx+ c$ , the line must first intersect the parabola: the point, (x, y), must satisfy $mx+n= ax^2+ bx+ c$ . But in order that it be tangent, that must be a double root which means that the discriminant of the equation $ax^2+ (b- m)x+ c- n= 0$ must be 0. The discriminant, $(b- m)^2- 4a(c- n)$ is quadratic in both b and m and so that equation may have two roots, while c, in $y= x^2+ x+ c$ is not quadratic.

Geometrically, increasing a, in the parabola, $y= x^2+ ax+ 1$ , moves the vertex to the left and downward. There will be two different values of a, one positive, the other negative, that will be tangent to y= x.
I have attached a graph.
Attachment 27084

For the second problem, y= mx+ 0.5 can be any line passing through (0, 0.5). That point is NOT on the given parabola so there exist two different lines through it that are tangent to the parabola, one above y= 0 and one below y= 0.
Attachment 27085

Here is a graph of the last one:
Attachment 27086