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Tangent line and normal on a parabola

  1. Aug 14, 2016 #1
    1. The problem statement, all variables and given/known data
    If the normal at P(ap^2 ,2ap) to the parabola y^2 = 4ax meets the curve again at Q(aq^2, 2aq), show that p^2 +pq+2=0

    2. Relevant equations
    Point-slope form

    3. The attempt at a solution
    Capture.jpg
    I tried putting y=2aq and x=aq^2 but I can seem to simplify the whole thing other than dividing both sides by a
     
  2. jcsd
  3. Aug 14, 2016 #2

    ehild

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    The expression is also dividable by q-p.
     
  4. Aug 14, 2016 #3
    Strangely enough I got this:
    Capture.png
     
  5. Aug 14, 2016 #4

    ehild

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    There should be minus in front of the last term in the first equation. You made a mistake when copying.
     
  6. Aug 14, 2016 #5
    Thanks - worked that out quickly.
     
  7. Aug 15, 2016 #6
    Now I'm stuck on the second part:

    Show that the equation of the locus of the point of intersection of the tangents at P and Q to the parabola is y^2(x+2a)+4a^3 =0. What does this mean? Does this mean that the tangents at P and Q meet at a point?

    I managed to find the points of intersection of the two tangents (apq, a(p+q)), but I can't seem to continue at this point.
     
  8. Aug 15, 2016 #7

    ehild

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    You have to work with x and y. What are p and q now?
     
  9. Aug 15, 2016 #8
    x=apq
    y=a(p+q)
     
  10. Aug 15, 2016 #9

    ehild

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    Have you copied the question correctly? These x and y values do not fulfill the equation given y^2(x+2a)+4a^3 =0.
     
  11. Aug 15, 2016 #10
    I'm sorry - but I can't catch it.
     
  12. Aug 16, 2016 #11

    ehild

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    I mean the problem might be wrong. The coordinates of the point of intersection do not fit to the given locus.
     
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