How Do Tangents Differ Between y²=4ax and x²=4ay?

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    Parabola Tangent
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Discussion Overview

The discussion revolves around the differences in the equations of tangents for the parabolas defined by y²=4ax and x²=4ay. Participants explore the derivation of tangent equations and the implications of the slope and intercept in these contexts, addressing potential confusion in terminology and notation.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant states that the tangent for the parabola y²=4ax is given by y=mx+(a/m), leading to c=a/m.
  • Another participant questions the meaning of "m" in the context of the tangent equation, suggesting that the initial explanation lacks clarity.
  • A participant asserts that if various values of m yield all tangents to the parabola, it would make sense, but they argue this is not the case.
  • It is clarified that m represents the slope of the tangent line, while c is the y-intercept.
  • One participant agrees with the previous point but acknowledges a mistake in their earlier graph check regarding the tangent derivation.
  • Another participant challenges the justification of a specific step in the derivation, indicating confusion over the roles of m and c in the tangent equations for both parabolas.
  • It is noted that m and c can have different interpretations depending on the context of the equations being discussed.

Areas of Agreement / Disagreement

Participants express differing views on the clarity of the tangent definitions and the roles of m and c in the equations. There is no consensus on the derivation steps or the implications of the tangent equations for the two parabolas.

Contextual Notes

There are unresolved issues regarding the definitions and interpretations of m and c in the context of different parabolic equations, which may lead to confusion in deriving tangents.

fireflies
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I know a tangent drawn on parabola having equation like
y^2=4ax is
y=mx+(a/m)
which provides c=a/m
Then how is it going to turn for equation like x^2=4ay?

From my derivation it will be like -c=am^2
when the equation of tangent is y=mx+c.

The derivation comes from the following:
y=mx+c
or, x=(y/m)-(c/m)

So, comparing with the tangent on
y^2=4ax we get

-(c/m)=a/(1/m)
that is -c= am^2

But the problem arises when in a question saying find the common tangent on y^2=4ax and x^2=4ay, the solution was made taking the tangents for each parabola as
y=mx+a/m and
x= my + a/m respectively.

Shouldn't the later one be x=(y/m)-(c/m)
i.e x=(y/m)-am ?
 
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What you say,
"I know a tangent drawn on parabola having equation like
y^2=4ax is
y=mx+(a/m)
which provides c=a/m"

Simply doesn't mean anything because you have not said what "m" means!

Please rewrite this, telling us what "m" is, so that it makes sense.
 
HallsofIvy said:
Simply doesn't mean anything because you have not said what "m" means!

Please rewrite this, telling us what "m" is, so that it makes sense.

If choosing various m gave all the tangents to the parabola, then it'd make sense. This is not the case however.
 
It's a general straight line equation, where m is the slope of a tangent
 
Mentallic said:
If choosing various m gave all the tangents to the parabola, then it'd make sense. This is not the case however.
Mentallic said:
If choosing various m gave all the tangents to the parabola, then it'd make sense. This is not the case however.

Yes, it is the case.

If the tangent is y=mx+c

where m=slope of the line
c=intercept it cuts on y axis

And it is a tangent to a general parabola of equation

y^2=4ax

then c=a/m
 
Yes, sorry, you're right. I made a quick base-case graph check and it went wrong somewhere along the way.

The derivation for your new tangent problem is very simple to derive. Since you've already correctly found that
y=mx+a/m is tangent to y^2=4ax
then symmetrically,
x=my+a/m is tangent to x^2=4ay
fireflies said:
So, comparing with the tangent on
y^2=4ax we get

-(c/m)=a/(1/m)
that is -c= am^2
You'll need to justify this last step. I can't follow that line of thought.
 
Last edited:
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Mentallic said:
You'll need to justify this last step. I can't follow that line of thought.

In this line you are taking x=my+c as a tangent of the x^2=4ay

Here, m is not the slope, c is not the intercept of y-axis.

Here m= 1/(slope) and
c=-(intercept/slope)

I just put in case of m, 1/m and in place of c, -(c/m) according to the conventional meaning of m and c.

I just tried it on paper. Both are actually same, denoting the same meaning until you are confused what m and c means in which equation.
 

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