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P is the point (2ap,ap^2) on the parabola x^2=4ay. The tangent at P meets the axis of the parabola at T and PN is drawn perpendicular to the axis, meeting it at N. The directrix meets the axis at A. (a) Prove OS=OA

Mathematics
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where is point S?
Um, idk, I think S is the focal length
the definition of the parabola is that the distance between the point P with the focus S is equal to the distance from the point P to the linear directrix. OS=OA is the special situation when P is point O.

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Other answers:

Is this what it asks?
idk, i don't understand the question, i just don't understand parametrics xD
Can you help ?
I don't know what the question is asking for.
You must provide more details.
Oh yeah I forgot sorry a) Prove (i) OS=OA (O is vertex, S is the focus)
then i have typed the answer: the definition of the parabola is that the distance between the point P with the focus S is equal to the distance from the point P to the linear directrix. OS=OA is the special situation when P is point O since point O is on the parabola.
Don't you have to solve it or something to prove that OS=OA like finding the equation of tangent
for the question a) OS=OA, I don't think so.
i think you will use it when you are going to solve other questions
Okay, I will draw the diagram that I drew can you check if its correct ?
no problem.
|dw:1318745104246:dw|
Is it right ?
uhh, actually, the directrix is Y=-a, not PN, so point A should be (0,-a) not (2ap, 0) and I don't know why there is another point P on the left side.
hmm..how does it look like then ? In my textbook, in one of the examples there's another P as well so I assume that there's another P in this question idk
for now, you don't need to worry about the other P, just to solve the question will be OK.
Okay, how can I solve it ?
which question? question a is solved through definition of parabola by using the special situation when the point P on the parabola moves to the vertex which is point O.
PS=PN, P is the point on the parabola----the definition
Okay thank you. But one more how can I prove ON=OT, do I use the same definition ?
OK, there is one more mistake which is the axis is X=0 not Y=0 since this parabola is vertical not horizontal. let me draw another diagram just to let you know what it should be. |dw:1318746374530:dw|
wow~ thank you. I drew it totally wrong *sigh* I just don't understand parametrics.
parabola can be vertical or horizontal which depends on the format of the function. The best way to differentiate which is which for me is to learn how to draw it correctly first.
Yeah, I should practice on it. How would I know if its vertical or horizontal ?
X^2=something*Y is vertical, Y^2=something*X is horizontal.
Oh I get it, thank you very much ^_^
You are welcome. I am a PHD student, so heyhey.
Cool~ PHD student. Wonder why you're smart.
Thank you. I am just happy to help people especially in Mathematics because I love it. Don't be afraid of it. It is really cool when you see the essence inside of it.
Lols, I'm afraid it esp parametrics, its like so~ HARD.
of it*
i see. Parametrics is harder but you can consider of them as changeable constants. They are constants but they are changing in different situation.
Oh cool~
back to the question ON=OT, you can not use definition since they are not distances in definition. You need to solve the tangent equation and find out point T. I found it. but they are not equivalent. ON=P^2, OT=(1-2a)*P^2.
Okay, I will work it out.

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