Results 1 to 3 of 3

Math Help - A Parabola Property

  1. #1
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,864
    Thanks
    744

    A Parabola Property


    Triangle PQR is inscribed in parabola y \,=\,ax^2.

    Tangents at P,Q,R intersect at P'\!,Q'\!,R', forming triangle P'Q'R'.

    Show that the areas of the two triangles are in the ratio 2:1.

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Soroban View Post
    Triangle PQR is inscribed in parabola y \,=\,ax^2.

    Tangents at P,Q,R intersect at P'\!,Q'\!,R', forming triangle P'Q'R'.

    Show that the areas of the two triangles are in the ratio 2:1.
    I'll do this for the parabola y^2=4ax since I'm more familiar with that. But the proof applies to any parabola.

    Take the three points to be (ar^2,2ar),\ (as^2,2as,\ (at^2,2at). The area of the triangle PQR is half the absolute value of \begin{vmatrix}ar^2&2ar&1\\ as^2&2as&1\\ at^2&2at&1\end{vmatrix}. Using elementary row operations and expanding down the right-hand column, this comes out to be

    \begin{vmatrix}ar^2&2ar&1\\ a(s^2-r^2)&2a(s-r)&0\\ a(t^2-r^2)&2a(-r)t&0\end{vmatrix} = 2a^2(s-r)(t-r)\begin{vmatrix}s+r&1\\t+r&1\end{vmatrix} = 2a^2(s-r)(t-r)(s-t).

    The tangent at R is x-ty+at^2 = 0. It meets the tangent at S at the point P' = (ast,a(s+t)). Similarly, Q' = (atr,a(t+r)) and R' = (ars,a(r+s)). The area of the triangle P'Q'R' is half the absolute value of

    \begin{vmatrix}ast&a(s+t)&1\\ atr&a(t+r)&1\\ ars&a(r+s)&1\end{vmatrix} = \begin{vmatrix}ast&a(s+t)&1\\ at(r-s)&a(r-s)&0\\ as(r-t)&a(r-t)&0\end{vmatrix} = a^2(r-s)(r-t)\begin{vmatrix}t&1\\s&1\end{vmatrix} = a^2(r-s)(r-t)(t-s).

    Thus area(PQR) = 2area(P'Q'R').
    Last edited by Opalg; December 28th 2010 at 02:19 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
    From
    Medgidia, Romania
    Posts
    1,252
    Thanks
    5
    Let P(x_1,ax_1^2), \ Q(x_2,ax_2^2), \ Q(x_3,ax_3^2)

    The slopes of the tangents at P, \ Q, \ R are:

    m_1=2ax_1, \ m_2=2ax_2, \ m_3=2ax_3

    The equations of the tangents are:

    y-ax_1^2=2ax_1(x-x_1)

    y-ax_2^2=2ax_1(x-x_2)

    y-ax_3^2=2ax_1(x-x_3)

    Solving the systems formed by two of three equations we find the coordinates of the points P', \ Q', \ R':

    P'\left(\dfrac{x_2+x_3}{2},ax_2x_3\right), \ Q'\left(\dfrac{x_1+x_3}{2},ax_1x_3\right), \ R'\left(\dfrac{x_1+x_2}{2},ax_1x_2\right)

    The area of the triangle P'Q'R' is A_{P'Q'R'}=\dfrac{1}{2}|\Delta '|

    where \Delta '=\begin{vmatrix}\dfrac{x_1+x_2}{2} & ax_1x_2 & 1\\<br />
\dfrac{x_2+x_3}{2} & ax_2x_3 & 1\\<br />
\dfrac{x_1+x_3}{2} & ax_1x_3 & 1\end{vmatrix}=\dfrac{a}{2}\begin{vmatrix}x_1+x_2 & x_1x_2 & 1\\<br />
x_2+x_3 & x_2x_3 & 1\\<br />
x_1+x_3 & x_1x_3 & 1\end{vmatrix}=\dfrac{a}{2}(x_1-x_2)(x_3-x_2)(x_3-x_1)

    Then A_{P'Q'R'}=\dfrac{1}{4}|a(x_1-x_2)(x_2-x_3)(x_3-x_1)|

    But A_{PQR}=\dfrac{1}{2}|\Delta|

    where \Delta=\begin{vmatrix}x_1 & ax_1^2 & 1\\<br />
x_2 & ax_2^2 & 1\\<br />
x_3 & ax_3^2 & 1\end{vmatrix}=a\begin{vmatrix}1 & 1 & 1\\x_1 & x_2 & x_3\\x_1^2 & x_2^2 & x_3^2\end{vmatrix}=a(x_3-x_2)(x_3-x_1)(x_2-x_1)

    Then A_{PQR}=\dfrac{1}{2}|a(x_1-x_2)(x_2-x_3)(x_3-x_1)|

    and A_{P'Q'R'}=\dfrac{1}{2}A_{PQR}
    Last edited by red_dog; December 28th 2010 at 12:40 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] gcd property
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: July 22nd 2011, 05:02 AM
  2. Mean Value Property Implies "Volume" Mean Value Property
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 16th 2011, 09:13 PM
  3. Another parabola property
    Posted in the Math Challenge Problems Forum
    Replies: 1
    Last Post: December 28th 2010, 09:00 AM
  4. Replies: 1
    Last Post: August 8th 2010, 12:22 AM
  5. Property Help
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: November 21st 2006, 03:27 PM

Search Tags


/mathhelpforum @mathhelpforum