Results 1 to 2 of 2

Thread: Parabola and Perpendicular Tangents

  1. #1
    Junior Member
    Joined
    Feb 2008
    From
    Istanbul
    Posts
    51

    Post Parabola and Perpendicular Tangents

    $\displaystyle f(x)=x^2-4x+n$ and the point which is not on the parabola (1,0) are given.If the tangent lines which drawn from (1,0) to parabola are perpendicular to each other, find n ? Thx for help
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Peritus's Avatar
    Joined
    Nov 2007
    Posts
    397
    the slope of the tangents to the parabola is given by:

    $\displaystyle \frac{d}
    {{dx}}f(x) = 2x - 4$

    let us denote the x coordinate (where it touches the parabola) of the first line as:$\displaystyle x_1$ and that of the second line as $\displaystyle x_2$, now we're told that the two tangents are perpendicular thus:

    $\displaystyle \left( {2x_1 - 4} \right)\left( {2x_2 - 4} \right) = - 1$

    the equations of the two lines are:

    $\displaystyle
    \left\{ \begin{gathered}
    y = \left( {2x_1 - 4} \right)\left( {x - x_1 } \right) + x_1 ^2 - 4x_1 + n \hfill \\
    y = \left( {2x_2 - 4} \right)\left( {x - x_2 } \right) + x_2 ^2 - 4x_2 + n \hfill \\
    \end{gathered} \right.
    $

    we know that the two lines intersect at (1,0), thus:

    $\displaystyle
    \left( {2x_1 - 4} \right)\left( {1 - x_1 } \right) + x_1 ^2 - 4x_1 + n = \left( {2x_2 - 4} \right)\left( {1 - x_2 } \right) + x_2 ^2 - 4x_2 + n
    $

    so we've got two equation in two unknowns $\displaystyle x_1, x_2$, find them and then finding n is trivial...
    Last edited by Peritus; Feb 15th 2008 at 11:10 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Tangents
    Posted in the Trigonometry Forum
    Replies: 4
    Last Post: Jul 18th 2011, 10:37 AM
  2. Replies: 2
    Last Post: Apr 4th 2009, 06:09 PM
  3. tangents
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Dec 15th 2008, 09:53 AM
  4. Tangents
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Feb 27th 2008, 06:36 PM
  5. Tangents
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Oct 29th 2007, 10:05 PM

Search Tags


/mathhelpforum @mathhelpforum