Parabola/Directrix/Focus

[Nazariy]

Hello friends,

Is there a way to show that given a directrix of y=k and focus of (a,b), the lengths from focus to point on parabola and from point on parabola to directrix are same without actually deriving the parabola equation in directrix/focus form?

i was thinking about expressing some variables in terms of others and then showing that LHS=RHS, therefore lengths are similar.. But I cannot find a way to express variables differently.

thank you

 

[pka]

Is there a way to show that given a directrix of y=k and focus of (a,b), the lengths from focus to point on parabola and from point on parabola to directrix are same without actually deriving the parabola equation in directrix/focus form?

THINK about what you are asking! One does not prove a definition.

 

[Nazariy]

THINK about what you are asking! One does not prove a definition.

Oh I see... I was just wondering why \(\displaystyle y^2=4ax\). Not 2a, not 5a, but 4a. That is why I started asking that question. But if you formulate the definition like so, then you will arrive at 4ax if your focus is at (a,0). Ok.

In that case why is this result useful? Why not consider such a "focus" point where length from point on parabola to directrix is 1/2 length from "focus" to that point (if at all possible). Why is the result where lengths are same more useful than any other result? (I am just thinking why/how would anyone come up with this directrix/focus idea)

 

[pka]

Why not consider such a "focus" point where length from point on parabola to directrix is 1/2 length from "focus" to that point (if at all possible). Why is the result where lengths are same more useful than any other result? (I am just thinking why/how would anyone come up with this directrix/focus idea)

You should always ask the question that you intend to ask.
Any parabola has an axis of symmetry, which contains the focus & vertex of the parabola.
Clearly the vertex must be half way between the directrix and the focus. See that webpage.

 

[Ishuda]

Oh I see... I was just wondering why \(\displaystyle y^2=4ax\). Not 2a, not 5a, but 4a. That is why I started asking that question. But if you formulate the definition like so, then you will arrive at 4ax if your focus is at (a,0). Ok.

In that case why is this result useful? Why not consider such a "focus" point where length from point on parabola to directrix is 1/2 length from "focus" to that point (if at all possible). Why is the result where lengths are same more useful than any other result? (I am just thinking why/how would anyone come up with this directrix/focus idea)

If you look at the ratio r of the lengths, the parabola is defined as r=1. Ellipses (includes circles) and hyperbolas are for r>1 and r<1 depending on just how you define the ratio. The those three define the set of conic sections, for example see Conic section - Wikipedia.

 

[Nazariy]

If you look at the ratio r of the lengths, the parabola is defined as r=1. Ellipses (includes circles) and hyperbolas are for r>1 and r<1 depending on just how you define the ratio. The those three define the set of conic sections, for example see Conic section - Wikipedia.

Thank you