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Math Help - Perpendicular Tangent Lines

  1. #1
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    Perpendicular Tangent Lines

    F(x) = kx^2 + 3

    If the tangent lines to the graph of F at (t, F(t)) and (-t, F(-t)) are perpendicular find t in terms of k.


    I don't know where to start really so if someone could walk me through this I'd greatly appreciate it.
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  2. #2
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    Definition of derivative of F(x) at a given point : It's the slope of the tangent to the graph of F(x) at that point.

    So find the derivatives at t and -t, to obtain the slope of the respective tangents at the two points.

    Now two lines are perpendicular if their slopes m1, m2 (say.) are such that (m1)(m2) = -1

    Use this to find t in terms of k.

    Note that you'll have to solve a quadratic equation in t.
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  3. #3
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    Quote Originally Posted by Naples View Post
    F(x) = kx^2 + 3

    If the tangent lines to the graph of F at (t, F(t)) and (-t, F(-t)) are perpendicular find t in terms of k.


    I don't know where to start really so if someone could walk me through this I'd greatly appreciate it.
    F'(x) = 2kx.


    F'(t) = 2kt

    F'(-t) = -2kt.


    Since the tangent lines are perpendicular,

    F'(t)\cdot F'(-t) = -1

    2kt \cdot (-2kt) = -1

    -4k^2t^2 = -1

    t^2 = \frac{1}{4k^2}

    t = \pm \frac{1}{2k}.
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