Need help with questions please

I have done the implicit differentiation required from part 6a) but I dont know where to start to complete part 6b)

There aren't really any shortcuts to finding distances - you need the coordinates of P and Q. That's the goal.
(Btw, it's a good idea to literally write down on your script "To find the distance, we wish to find the coordinates of P and Q". Probably this sentence worth no marks, but it's good for the marker, but more importantly for you.)

Let's start with finding P, as finding Q is practically the same. You know:
(i) the tangent to the curve at P is parallel to the y-axis. How does this relate to dy/dx (recall dy/dx is just the gradient of the tangent line)?
(ii) P lies on the curve. Using (i), can you set up an equation to find the coordinates of P?
(Again, it's a good idea to have some descriptive words like "since the tangent at P is parallel to the y-axis, we have 'blah blah blah and calculations'")

Do you happen to have an answer as to what k is? If my long-winded method gives the correct answer, then I'll explain what I did, but I don't know if it's even right.

K=9/4

On the mark scheme they split up the gradient for example to find P they use 4x-16y=0. I dont understand why they do this

Do you happen to have an answer as to what k is? If my long-winded method gives the correct answer, then I'll explain what I did, but I don't know if it's even right.

On the mark scheme they split up the gradient for example to find P they use 4x-16y=0. I dont understand why they do this

This is because the gradient is parallel to the y-axis ie infinite.

Ok thanks. But why do these only choose the denominator of the gradient?

Anything/0 is infinte - the denominator comes into play for the other point.

What other point?

You must not give a full solutions as that will break the rules. The poster above has given a hint.

What other point?

You need P and Q to get the distance between them.

When you make the denominator of the derivative 0, you get 4x = 16y which is easy enough to sub into the curve's original equation to get a coordinate for P. However, I'm struggling to apply the same method for Q by making 4x - 4y - 9 equal 0 since putting this into the curve's original equation is difficult, and seems to give me impossible/convoluted values for the coordinates of Q. Is there something I'm doing wrong?

There are 8 marks - can you post your working?

I've just been scribbling values I'm getting on paper whilst using an online calculator (lost mine so I need to order a new one).

For Q:

4x - 4y - 9 = 0
4x = 4y + 9
Original equation of curve: 4xy = 2x^2 + 8y^2 - 36y
(4y + 9)y = 2(4y+9/4)^2 + 8y^2 - 9(4y+9/4)
4y^2 + 9y = (16y^2 + 81)/8 + 8y^2 - (36y + 81)/4
I multiplied everything by 8 here to get rid of the fractions
32y^2 + 72y = 16y^2 + 81 + 64y^2 - (72y + 162)
Collected like terms
48y^2 - 144y - 81 = 0
This is the quadratic equation that I'm left with, and it gives me wacky coordinates.

When applying the same process for P, I just get (6, 1.5) which is a normal set of coordinates, so I'm unsure why the same doesn't work for Q.

Id have noted that part of the expression is
4y^2 - 4xy = y(4y-4x) = -9y
which cuts down on work/errors. Then ...