Parametrics proof (1 Viewer)

[byakuya kuchiki]

P(2ap, ap^2) and Q(2aq, aq^2) are two points on the parabola x^2 = 4ay. The tangent at P and the line through Q parallel to the x axis of the parabola meet at point R.
The tangent at Q and the line through P parallel to the axis of the parabola meet at point S.

Show the co-ordinates of S and R are (2ap, 2apq-aq^2) and (2aq, 2apq-ap^2) respectively.

 

[Trebla]

P(2ap, ap^2) and Q(2aq, aq^2) are two points on the parabola x^2 = 4ay. The tangent at P and the line through Q parallel to the x axis of the parabola meet at point R.
The tangent at Q and the line through P parallel to the axis of the parabola meet at point S.

Show the co-ordinates of S and R are (2ap, 2apq-aq^2) and (2aq, 2apq-ap^2) respectively.

The tangent at P has equation (which is easy to derive):
y = px - ap²
The tangent at Q has equation similarly:
y = px - aq²
The line through P parallel to the axis of the parabola is x = 2ap
The line through Q parallel to the axis of the parabola is x = 2aq

To find the y coordinate of R sub x = 2aq into equation of tangent at P:
y = 2apq - ap²

To find the y coordinate of S sub x = 2ap into equation of tangent at P:
y = 2apq - aq²