Find the equation of the tangent of the parabola y^2= 4px

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Discussion Overview

The discussion revolves around finding the equations of tangents to specific parabolas, specifically \(y^2 = 4px\) and \(Y^2 = 10x\). The participants are exploring methods to solve these problems using analytic geometry rather than calculus, addressing the conditions for tangency and the intersection of tangents at specific points.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant requests help with finding the tangent to the parabola \(y^2 = 4px\) that is perpendicular to a given line, and also seeks the tangent to \(Y^2 = 10x\) at the latus rectum's extremities.
  • Another participant suggests finding the gradient of the given line and using it to determine the points on the parabola with the same gradient, implying the use of derivatives.
  • A different participant clarifies that they cannot use calculus methods and must rely solely on analytic geometry.
  • One participant provides a detailed approach to the first question, explaining the relationship between the slope of the tangent line and the conditions for tangency, including the use of the discriminant for determining double roots.
  • The same participant describes the latus rectum of the second parabola and outlines how to find the equations of the tangents at the endpoints, emphasizing the need for double roots in the resulting equations.

Areas of Agreement / Disagreement

There is no consensus on the methods to be used, as some participants propose calculus-based approaches while others insist on using only analytic geometry. The discussion remains unresolved regarding the best approach to solve the problems presented.

Contextual Notes

Participants express limitations in their methods, particularly the restriction against using calculus, which may affect the completeness of their solutions. The discussion also highlights dependencies on the definitions of terms like "latus rectum" and conditions for tangency.

callas
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hello everyone,
:confused:i'm having can't seem to solve these two questions...

1) Find the equation of the tangent of the parabola y^2= 4px, perpendicular to the lines 4Y- X + 3=0, and find the point of contact?

2) Find the equation of the tangent to the parabola Y^2= 10x, at the extremities of the lactus rectum. On what line do these tangents intersect?

If there is anyone out there who knows the answer...could u please explain or direct me to a suitable site which may explain the concepts.:cry:


Thanks callas
 
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1) Can you find the gradient of that line? It says to find the tangent parallel to that line so when you take the derivative of the parabola, you're going to have to make it equal to the gradient of the line to find the value(s) of the points on the parabola that are have this gradient.

2) Do you know how to find the equation of the latus rectum on a parabola? You'll have to again take the derivative of the parabola and then find what the gradients of the tangents are at the points where the latus rectum intersects the parabola. Since you'll know the it should be fairly simple to find the point of intersection between the tangent and the parabola, so you should be able to form an equation for each tangent. Now just find where these tangents intersect and notice the point's similarity to the focus and latus rectum.
 


thanks...but i can not use any calculus method...it must only be solved using analytic geometry...yes i could find the gradient...
 


callas said:
hello everyone,
:confused:i'm having can't seem to solve these two questions...

1) Find the equation of the tangent of the parabola y^2= 4px, perpendicular to the lines 4Y- X + 3=0, and find the point of contact?
Any line perpendicular to 4y- x+ 3= 0 must have slope -4 and is of the form y= -4x+ a or x= -(1/4)y+ b. That line will cross to y^2= 4px or x= (1/4p)y^2 where x= -(1/4)y+ b= y^2/4p and will be tangent if and only if that equation has a double root. That is equivalent to the quadratic equation y^2+ py- 4pb= 0 and has a double root if and only if the discriminant, p^2- 4(-4pb)= p^2+ 16pb= 0. Solve that for b.

2) Find the equation of the tangent to the parabola Y^2= 10x, at the extremities of the lactus rectum. On what line do these tangents intersect?
The "latus rectum" of this parabola is the vertical line segment, passing through the focus of the parabola with endpoints on the parabola. This particular parabola has focus at (2.5, 0) and its endpoints are where y^2= 10(2.5) or (2.5, -5) and (2.5, 5).
Any line passing through (2.5, -5) has the form x= m(y+5)+ 2.5. Again, use the fact that we must have a double root for x= (1/10)y^2= m(y+5)+ 2.5, which is equivalent to y^2- 10my+ 7= 0 to find m. Do the same with (2.5, 5).

If there is anyone out there who knows the answer...could u please explain or direct me to a suitable site which may explain the concepts.:cry:


Thanks callas
 

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