Parametrics past questions

[spnmox]

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

[fun_jirachi]

Hey there!

For the first one, I assume you understand and did the question? If not, ask again. QT is a diameter if and only if P is the point (0, 0). By the way, it doesn't actually mention that QT is a diameter, or make any mention of T at all (I'm a bit confused here.)

For the second one, you can just equate the gradients. The logic here is that R is defined as being on the normal at T on P1. Therefore, if the normal at R to P2 has the same gradient as the normal at T to P1, it's automatically part of the same line because R was defined to be in the question.

Ask more if you need, hope this helps

[spnmox]

hey! thanks so much, you helped with the second one and i just asked my teacher for help with the first one