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

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In summary, we discussed the condition of tangency for a line and a parabola, where the line must cut the parabola at one point and the discriminant of the resulting equation must be equal to zero. However, this condition only applies when the slope of the line, m, is not equal to zero. This means that the equation is a quadratic and the line is tangent to the parabola.
  • #1
vkash
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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 let's 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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  • #2
vkash said:
[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
Citan Uzuki said:
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
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.
 
  • #5


As a scientist, it is important to carefully review and analyze any proof or evidence presented. In this case, it seems that there may be a flaw in the proof, as the example of y=2 does not follow the condition that was previously proved. It is possible that there is a mistake in the algebraic calculations or in the initial assumptions made. It would be helpful to review the proof and check for any errors or inconsistencies. Additionally, it may be useful to consult other sources or experts in the field to see if there is a different approach or explanation for the condition of tangency in parabolas. Overall, it is important to remain critical and open-minded when evaluating scientific evidence and theories.
 

1. What could be causing the discrepancy between the proof and my understanding?

There could be a number of factors that contribute to this issue. It's possible that there is an error in the proof itself, or that there is a flaw in your understanding of the concept being proven. It's also possible that there are external factors, such as missing information or incorrect assumptions, that are affecting your understanding.

2. How can I determine whether the problem lies with the proof or with my understanding?

One way to determine this is to carefully examine the proof and try to identify any errors or inconsistencies. You can also consult with other experts in the field to get their opinions and insights. Additionally, you can try to explain the concept to someone else and see if they understand it in the same way as you do.

3. Should I try to solve the problem on my own or seek help from others?

It's always a good idea to try to work through the problem on your own first, as this can help you develop your critical thinking and problem-solving skills. However, if you're struggling to find a solution, don't hesitate to seek help from others, whether it be a colleague, mentor, or online community.

4. Is it common to encounter discrepancies between proofs and understanding in the scientific community?

Yes, it is quite common for scientists and researchers to encounter discrepancies between proofs and their understanding. This is a natural part of the scientific process and can often lead to new discoveries and advancements in the field.

5. What steps can I take to prevent or reduce the occurrence of discrepancies in my work?

To reduce the chances of encountering discrepancies, it is important to have a thorough understanding of the concepts and principles involved in your research. It's also important to carefully check and double-check your work, as well as seek feedback from others. Additionally, staying up-to-date with the latest research and developments in your field can help prevent discrepancies from arising.

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