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Tangent line and normal on a parabola

  • #1
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Homework Statement


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

Homework Equations


Point-slope form

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
 
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Answers and Replies

  • #2
ehild
Homework Helper
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Homework Statement


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

Homework Equations


Point-slope form

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
The expression is also dividable by q-p.
 
  • #3
173
2
Strangely enough I got this:
Capture.png
 
  • #4
ehild
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Strangely enough I got this:
Capture.png

There should be minus in front of the last term in the first equation. You made a mistake when copying.
 
  • #5
173
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Thanks - worked that out quickly.
 
  • #6
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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.
 
  • #7
ehild
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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.
You have to work with x and y. What are p and q now?
 
  • #8
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x=apq
y=a(p+q)
 
  • #9
ehild
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x=apq
y=a(p+q)
Have you copied the question correctly? These x and y values do not fulfill the equation given y^2(x+2a)+4a^3 =0.
 
  • #10
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I'm sorry - but I can't catch it.
 
  • #11
ehild
Homework Helper
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I'm sorry - but I can't catch it.
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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