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Two values for m for which a line is tangent to a parabola

  1. Jul 20, 2014 #1
    1. The problem statement, all variables and given/known data

    Hi! I am doing some problems to practice for a math competition, and I'm wondering if I did this correctly. I don't really have an answer sheet, so I have no way of knowing whether I'm right. If you would please review it, that would be cool!

    It reads:

    There are exactly two values of m for which the line y=mx-12 is tangent to the parabola y=x^2-3x+5. Determine those two values.

    2. Relevant equations

    y=mx-12
    y=x^2-3x+5

    3. The attempt at a solution

    Okay so first I found the derivative of y=x^2-3x+5.

    y'=2x-3

    This is the slope, so I put it into the equation of the tangent line:

    y=mx-12=(2x-3)x-12=2x^2-3x-12 <-- the tangent line

    Now I made the the two equations equal to each other because they intersect at one point, right?

    so:

    x^2-3x+5=2x^2-3x-12

    and when I solve for x it gives me +/- √17

    I put this into the equation for the slope and I get:

    m= 2√17 -3 or m=-2√17 -3

    Is this correct? Thank you in advance.
     
  2. jcsd
  3. Jul 20, 2014 #2

    Nathanael

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    Homework Helper

    Looks correct to me
     
  4. Jul 21, 2014 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Another way to do it (used by Fermat previous to the development of calculus):
    To find where the line y= mx- 12 crosses the parabola [itex]y= x^2- 2x+ 5[/itex], solve the equation [itex]mx- 12= x^2- 2x+ 5[/itex] or [itex]x^2- (m+ 3)x+ 17= 0[/itex]

    The "discriminant" of that equation is [itex](m+3)^2- 4(1)(17)= m^2+ 6m+ 9- 68= m^2+ 6x- 59[/itex]. The line will not touch the parabola at all if the equation has no real roots- if the discriminant is negative. It will cross through the parabola if the equation has two real roots- if the discriminant is positive. It will touch the parabola once (be tangent to the parabola) if the equation has a single root- if the discriminant is zero.

    Thus, the line y= mx- 12 will be tangent to the parabola [itex]y= x^2- 2x+ 5[/itex] if and only if [itex]m^2+ 6m- 59= 0[/itex]. That is the same as the equation you got and, yes, it has roots [itex]-3\pm 2\sqrt{17}[/itex].
     
  5. Jul 21, 2014 #4

    Thanks!
     
  6. Jul 21, 2014 #5

    Ouuuu interesting. Thank you :)
     
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