Radius of Curvature: Parabola y^2=4ax

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Discussion Overview

The discussion revolves around determining the radius of curvature of the parabola defined by the equation y²=4ax at the end of a focal chord. Participants explore the mathematical derivation and clarify terminology related to focal chords and the latus rectum.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks for the radius of curvature at the end of a focal chord of the parabola y²=4ax.
  • Another participant requests additional context about the problem, suggesting that clarification on the specific focal chord is needed.
  • A participant presents the formula for radius of curvature and begins to derive it using the given parabola equation, calculating derivatives and expressing the radius in terms of y.
  • There is uncertainty about the definition of "focal chord," with one participant suggesting it might refer to the latus rectum, while another expresses confusion about the term itself.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definition of "focal chord" and whether it refers to the latus rectum. The discussion remains unresolved regarding the specific point at which the radius of curvature is to be calculated.

Contextual Notes

There are limitations in the discussion regarding the assumptions about the focal chord and the lack of clarity on how to identify the specific point on the parabola for the radius of curvature calculation.

amaresh92
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what is the radius of curvature of a parabola y^2=4ax at the end of the focal chord ?
 
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hi amaresh92! :wink:

show us what you've done, and where you're stuck, and then we'll know how to help! :smile:

(and which focal chord? do you mean the latus rectum? :confused:)
 
we got the formula,
r=[{(1+(dy/dx)^2)}^3/2]/(d2x/dy2)
we have,
y^2=4ax

2ydy/dx=4a
dy/dx=4a/2y=2/y

d2y/dx2=-2/y^2*dy/dx

d2y/dx2 =-2/y^2*2/y

d2y/dx2 =-4/y^3

r={(1+4/y^2)^3/2}/(-4/y^3)

then how to find the point at which it is asked?
its not a latus rectum i guess as it is mention only focal chord
 
amaresh92 said:
then how to find the point at which it is asked?
its not a latus rectum i guess as it is mention only focal chord

i've never heard of the term "focal chord" …

on that face of it, it should mean any chord through the focus …

so I'm guessing that they mean the latus rectum :confused:
 

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