Thread: Geometric significance

Find the values of p such that y=px intersects y=x^2 + 1 twice. I was able to do this fine, but the next part asked to explain the geometric significance of this. What is the geometric significance?

Hello, usagi_killer!

Find the values of such that intersects twice.

Explain the geometric significance of this.

The parabola and line intersect: .

. . Quadratic Formula: .

With two intersections, the discriminant must be positive:
. .

Hence: .


Make a sketch.
We have an up-opening parabola with vertex (0,1)
. . and a straight line through the origin.

Code:

        *       |       *
                |
         *      |      *
          *     |     *
            *   |   *     o
               1*       o
                |     o
                |   o
                | o
      - - - - - + - - - - - - - 
              o |
            o   |
                |

If the slope is +2 or -2, the line is tangent to the parabola.
. . There is one intersection.

If is greater than -2 and less than 2, the line misses the parabola.
. . There is no intersection.

If is less than -2 or greater than 2, there are two intersections.