Solving Two Parabola Problems: Proving & Finding Values

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Discussion Overview

The discussion revolves around two problems related to parabolas, focusing on proving a geometric property and finding specific values related to tangents and normals. The scope includes mathematical reasoning and problem-solving techniques in the context of conic sections.

Discussion Character

  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant presents a problem involving proving that the normal to the parabola \(y^2=4ax\) at a specific point intersects the parabola again at a certain angle, expressing difficulty in finding the solution.
  • The same participant poses a second problem about determining values of \(a\) for which tangents from a point not on the y-axis to one parabola are normals to another parabola, indicating uncertainty in how to approach it.
  • Another participant suggests starting with the first problem and encourages showing detailed working steps, proposing that reversing the roles of the x and y axes might be helpful.
  • This participant also emphasizes the importance of rewriting the second problem mathematically, specifying the conditions for the point from which tangents are drawn.
  • A later reply indicates that the first problem has been resolved by the original poster, who acknowledges having overcomplicated it.
  • There is a request for further discussion on the second problem, reiterating the earlier suggestions for clarity and detail in the approach.

Areas of Agreement / Disagreement

The discussion shows a mix of uncertainty and progress, with one participant claiming to have solved the first problem while the second problem remains unresolved and open for further exploration.

Contextual Notes

Participants have not provided complete solutions or detailed steps for the second problem, and there are unresolved assumptions regarding the conditions of the points involved in both problems.

varunKanpur
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I am not able to solve the following problem

#1) Prove that the normal to parabola y2=4ax at (am2,-2am) intersects the parabola again at an angle tan-1(m/2)

What I am thinking is to solve the equation of parabola and equation of normal y=mx-am-3-2am simultaneously and at that point I will find the slope of tangent and will get the angle between tangent and normal. The problem is that answer is not coming.

#2) For what values of a will the tangents drawn to parabola y2=4ax from a point , not on the y-axis, will be normals to the parabola x2=4y?

I have no idea on how to solve this question
Thanks in Advance
 
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Start with the first one to begin with.

Please show your working?
Show each step with your reasoning.
It can hep to reverse the roles of the x and y axes.

The key to the second one is to rewrite the problem statement in maths.
i.e. a point not on the y-axis is point ##p=(p_x,p_y): p_x\neq 0##
 
I got the #1 problem, I was making it more lengthy.
 
OK - so what about #2?
Same suggestions.
 

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