Proof of parabola equation

[Nazariy]

Hello!

I have a question regarding a parabola.

Suppose we construct a parabola, so that our vertex is at the origin of a coordinate plane and its directrix line is parallel to x-axis, also suppose our focus point has coordinates (0,p) and a point on the parabola P(x,y). How can we show that the equation of parabola is x^2=4py ?

 

[Nazariy]

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There is the graph for given parabola.

 

[pka]

Suppose we construct a parabola, so that our vertex is at the origin of a coordinate plane and its directrix line is parallel to x-axis, also suppose our focus point has coordinates (0,p) and a point on the parabola P(x,y). How can we show that the equation of parabola is x^2=4py ?

By definition the parabola is set of points equally distance from its directrix and its focus.

Thus if the directrix is \(\displaystyle y=-p\) then \(\displaystyle \sqrt{(y+p)^2}=\sqrt{x^2+(y-p)^2}\). Square both sides.

 

[Nazariy]

By definition the parabola is set of points equally distance from its directrix and its focus.

Thus if the directrix is \(\displaystyle y=-p\) then \(\displaystyle \sqrt{(y+p)^2}=\sqrt{x^2+(y-p)^2}\). Square both sides.

Was about to write that my feeling of utter awe has been swiftly replaced by me feeling silly when I did not get the above, but no I found my mistake.... swapped coordinates for F, used (p,0) not (0,p). Now I get the same result. Ahhh this is beautiful!! Thank you!!!!