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Math Help - Points of tangency.

  1. #1
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    Points of tangency.

    Two tangents to y=x^2-2x-3 pass through (5,3). Find the points of tangency.
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  2. #2
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    Quote Originally Posted by Hibijibi View Post
    Two tangents to y=x^2-2x-3 pass through (5,3). Find the points of tangency.
    Let the tangent to the curve be at (a, b). Note that b = a^2 - 2a - 3.

    The gradient m of the tangent is found by getting the value of the derivative at x = a:

    dy/dx = 2x - 2 therefore m = 2a - 2.

    So the equation of the tangent is

    y - b = m(x - a)

    => y - a^2 + 2a + 3 = (2a - 2)(x - a)

    => y = (2a - 2)x - 2a^2 + 2a + a^2 - 2a - 3 = (2a - 2)x - a^2 - 3.

    But the point (5, 3) lies on this tangent.

    Therefore 3 = (2a - 2)(5) - a^2 - 3

    => 3 = 10a - 10 - a^2 - 3

    => a^2 - 10a + 16 = 0

    => (a - 2)(a - 8) = 0

    => a = 2, 8.

    Substitute into b = a^2 - 2a - 3 to get the corresponding values of b.
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    Thanks! Really appreciate the time taken to help an uneducated sap such as myself.
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