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Math Help - find tangent quation

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    find tangent quation

    find tangent equation ( y=x^2 and y=x^2-4x+5) have common tangent.
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    Quote Originally Posted by hoger View Post
    find tangent equation ( y=x^2 and y=x^2-4x+5) have common tangent.
    1. Choose an arbitrary point on the first parabola: T(t, t^2)

    2. Calculate the equation of the tangent in T at the first parabola:

    y-t^2=2t(x-t)~\implies~y=2tx-t^2

    3. Calculate the point of intersection ofthe tangent and the second parabola:

    2tx-t^2 = x^2-4x+5~\implies~x = (2+t)\pm\sqrt{4+4t+t^2-5-t^2}

    4. The tangent of the first parabola is tangent to the second parabola too if the radical equals zero:

    4t-1=0~\implies~t=\dfrac14

    5. Therefore the equation of the common tangent is:

    y = \frac12 x-\frac1{16}
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