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It seems that something is wrong either with this proof or my understanding.

  1. Oct 23, 2011 #1
    Here i am asking all the things about parabola. I think something is wrong with proof where we prove condition of tangency.
    It is done shown(in my book)
    [itex]y^2=4ax[/itex] //equation of general parabola
    let us assume that line [itex]y=mx+c[/itex] is passing through parabola.
    points where it will cut parabola.
    [itex](mx+c)^2=4ax[/itex]
    solving this equation.
    [itex](mx)^2+x(2mc-4a)+c^2=0[/itex]
    line will touch parabola at one point if of the roots of this equation are equal.
    It means it’s discriminant is zero.
    [itex](2mc-4a)^2-4m^2c^2=0[/itex]
    => [itex]16a^2=16amc[/itex]
    [itex]a=0 [/itex]; not possible because in that case it will not remain a parabola it became a line y=0.
    [itex]a=mc[/itex]
    this is the require condition for the line to cut the parabola at one point.
    So lets take an example.
    [itex]y=2[/itex]. this line cuts the parabola [itex]y^2-4x=0[/itex] at one point. as we can see on the graphs
    but does it obey the equation proved previously.
    a=0/2. NO. it is not obeying that equation.
    WHY????????????????????????????????????????
    the condition is for line to cut the parabola at one point.y=2 is also a line that cuts parabola at one point but not obeying the condition.
     
    Last edited: Oct 23, 2011
  2. jcsd
  3. Oct 23, 2011 #2
    To conclude that this equation is a quadratic you need to know that m≠0. If m=0, then this is just a linear equation and will have exactly one solution regardless of what the discriminant is. Since your given line does in fact have m=0, the discriminant need not be zero, and so a is not necessarily equal to mc.
     
  4. Oct 24, 2011 #3
    great!!
    problem solved.
    million thanks for answering.
     
  5. Oct 24, 2011 #4

    HallsofIvy

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    By the way, when m is not 0, "touching the parabola at one point" means "tangent to the parabola" which is what the discriminant being 0 gives.
     
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