Hey! I was wondering if I could get some help on a couple of parametrics questions:
EXT 1 2014 HSC 13c
The point P(2at, at^2) lies on the parabola x^2=4ay with focus S. The point Q
divides PS internally in the ratio t^2:1. Using the result: mOQ=t, or otherwise, show that Q lies on a fixed circle of radius a.
I've had a look at the solutions but I don't understand how QT is a diameter (where T is 0,2a).
EXT 1 2016 HSC 14c
The point T(2at, at^2) lies on the parabola P, with the equation x^2=4ay. The tangent to the parabola P1 at T meets the directrix at D. The normal to the parabola P1 at T meets the vertical line through D at the point R.
iv) It can be shown that the minimum distance between R and T occurs when the normal to P1 at T is also the normal to P2 at R. (do not prove this) Find the values of t so that the distance between R and T is a minimum.
The question actually includes a diagram so I'm sorry if the question doesn't make as much sense on its own. Part i) required you to find the coordinates of point D, part ii) to show that the locus of R lies on another parabola P2, part iii) to state the focal length of the parabola P2.
According to the solutions, you just have to equate the gradients of the two normals to find t, but I don't really understand why you can just do that. Don't you have to equate the two equations, not just the gradients?
Any help is appreciated