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Math Help - 4 unknowns

  1. #1
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    4 unknowns

    Sketch the graphs of the two equations y=x^2 and y= -x^2+6x-5, and sketch the two lines that are tangent to both graphs. Find the equations of these lines.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    Actually there are only 2 unknowns

    the tangent lines at (a,f(a)) and (b,f(b) respectively are:

    1)y= 2a (x-a) + a^2

    2) y= (-2b+6)(x-b) - b^2 + 6b -5

    The slopes are the same so 2a = -2b + 6

    or a= -b+3

    Also if the lines tangent at (a,f(a)) and (b,f(b) are the same then 1) must intersect 2) at (b,f(b)

    So -b^2 + 6b - 5 = 2a (b-a) + a^2

    so we have the system:

    3) a= -b+3

    4) -b^2 + 6b - 5 = 2a (b-a) + a^2

    using 3) in 4)


    -b^2 + 6b -5 = 2(-b+3)(b- (-b+3)) + (-b+3)^2

    expand and solve

    See attachment
    Attached Thumbnails Attached Thumbnails 4 unknowns-tanlines2.jpg  
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