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

  • Thread starter vkash
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  • #1
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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.
 
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Answers and Replies

  • #2
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[itex](mx)^2+x(2mc-4a)+c^2=0[/itex]
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.
 
  • #3
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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.
great!!
problem solved.
million thanks for answering.
 
  • #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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