Another parabola property

**Soroban** · December 28th 2010, 06:55 AM

Point $\,P$ is on the parabola $y\,=\,x^2.$

The origin is: . $O(0,0).$

The perpendicular bisector of chord $\,OP$ has $\,y$ -intercept $\,b.$

$\displaystyle \text{Find: }\,\lim_{P\to O} b$

Code:


              |
             bo            ♥
              |\
              | \
              |  \        ♥P
              |   \     *
   ♥          |    \  *  ♥
    ♥         |     o   ♥
      ♥       |   *   ♥
         ♥    | *  ♥
  - - - - - - ♥ - - - - - - -
              O

**emakarov** · December 28th 2010, 08:00 AM

Let P have coordinates $(x_0,x_0^2)$ . Then the center of OP has coordinates $(x_0/2,x_0^2/2)$ . The slope of the chord is $x_0$ , so the slope of the perpendicular is $-1/x_0$ . Thus, the equation of the perpendicular bisector is $y=-x/x_0+b$ , and $y(x_0/2)=x_0^2/2$ . From here, $b=(x_0^2+1)/2$ . Therefore, $\lim_{P\to O}b=1/2$ and not $\infty$ as I thought initially.

I found the following theorem in this PDF document.

The perpendicular bisector of a parabola chord and the perpendicular to the axis through the center of the chord cut off a segment on the axis of length equal to the distance of the focus to the directrix.

The axis is the vertical one. Indeed, the length of the segment in question is $b-x_0^2/2=1/2$ , the focus of the parabola $y=x^2$ is $(0,1/4)$ and the directrix is $y=-1/4$ .

Thread: Another parabola property

LinkBack

Thread Tools

Display

Another parabola property

Similar Math Help Forum Discussions

[SOLVED] gcd property

Mean Value Property Implies "Volume" Mean Value Property

A Parabola Property

[SOLVED] Derivation of the Parabola's Focal Property Using a Sphere Withing a Cone

Property Help

Search Tags