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Math Help - Geometric significance

  1. #1
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    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?
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  2. #2
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    Hello, usagi_killer!

    Find the values of p such that y=px intersects y=x^2 + 1 twice.

    Explain the geometric significance of this.
    The parabola and line intersect: . x^2+1 \:=\:px \quad\Rightarrow\quad x^2 - px + 1 \:=\:0

    . . Quadratic Formula: . x \:=\:\frac{p \pm \sqrt{p^2-4}}{2}

    With two intersections, the discriminant must be positive:
    . . p^2 - 4 \:>\:0 \quad\Rightarrow\quad p^2 \:>\:4 \quad\Rightarrow\quad |p| \:>\:2

    Hence: . p \:<\:-2\;\text{ or }\;p \:>\:2


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

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

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


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