Parabola Midpoint: Find Equation of Locus

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SUMMARY

The discussion centers on finding the equation of the locus of the midpoint of segment PQ, where P is a point on the parabola defined by the equation y² = 4ax and Q is the intersection of the tangent at P with the y-axis. The user initially concluded that the locus is represented by 2y² = 9ax, while the correct answer is y² = 9ax. The discrepancy highlights the importance of verifying calculations in conic sections, particularly in relation to parabolic equations.

PREREQUISITES
  • Understanding of parabolic equations, specifically y² = 4ax
  • Knowledge of locus concepts in coordinate geometry
  • Familiarity with tangent lines and their properties
  • Basic algebra for manipulating equations
NEXT STEPS
  • Study the derivation of the locus of points related to conic sections
  • Explore the properties of tangents to parabolas
  • Learn about midpoint formulas in coordinate geometry
  • Review examples of similar problems involving parabolas and loci
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Students studying conic sections, geometry enthusiasts, and educators looking to clarify concepts related to parabolas and their properties.

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Homework Statement



A variable tangent at P to the parabola y2=4ax meets the y-axis at Q. Find the equation of the locus of the midpoint of PQ. As shown below, my final answer is 2y2=9ax but the answer provided is y2=9ax. Can anyone correct me?

Homework Equations


The Attempt at a Solution



DSC00596 - Copy.jpg
 
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Michael_Light said:

Homework Statement



A variable tangent at P to the parabola y2=4ax meets the y-axis at Q. Find the equation of the locus of the midpoint of PQ.


As shown below, my final answer is 2y2=9ax but the answer provided is y2=9ax. Can anyone correct me?

Homework Equations





The Attempt at a Solution



View attachment 34238

Might I surgest you take look at this website here.

http://home.scarlet.be/~ping1339/parabola.htm

It deals with a very simular problem :)

/Fred
 
matematikeren said:
Might I surgest you take look at this website here.

http://home.scarlet.be/~ping1339/parabola.htm

It deals with a very simular problem :)

/Fred

Its a very good website... but would you kindly help me with my question first?? ><''
 
Michael_Light said:

Homework Statement



A variable tangent at P to the parabola y2=4ax meets the y-axis at Q. Find the equation of the locus of the midpoint of PQ.


As shown below, my final answer is 2y2=9ax but the answer provided is y2=9ax. Can anyone correct me?

Homework Equations





The Attempt at a Solution



View attachment 34238

If it's any help to you, I worked it myself without looking at what you did and I got the same answer you did.
 

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