Help with parabola questions~! (1 Viewer)

[peterpandied]

Q. 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.

Prove: (i) OS = OA (O is vertex, S the focus)
(ii) ON = OT
(iii) the x axis bisects PT at a point, say B



I know they're right but I just have no idea on proving it~ >.<
plz help~

 

[Mountain.Dew]

Q. 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.

Prove: (i) OS = OA (O is vertex, S the focus)
(ii) ON = OT
(iii) the x axis bisects PT at a point, say B

I know they're right but I just have no idea on proving it~ >.<
plz help~

i have a tendency to give away the answers, but this time i will only explain what to do.

for part (i) this one is simple. knowing that S is pt (0,a) and the directrix is y = -a, show the DISTANCE OA is same distance as OS.
(ii) find pt N, and find pt T. N is simply (0, 2ap). for T, find equ of tangent at P, then sub x=0 to find y-coordinate of T. then, show ON = OT
(iii) for this one, u have pt P and pt T. lets say M is the midpt of PT. PT INTERSECTS at the x-axis, B. SO, in equ of tangent at P, let y =0 to find x-coordinate of B. Then, show that B has the same coordinates as M.

Thats my 2 cents...

 

[peterpandied]

ahh~! thx ~!

 

[Mountain.Dew]

it was a pleasure. always willing to help.