1. ## 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?

2. Hello, usagi_killer!

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

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

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

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

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

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

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

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

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### geomentic significace

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