Hi! Very rusty with locus and param, need someone to help with these
1) Find the gradients of the common tangents of the parabola y= x(x-4) and the circle (x-2)^2 + y^2 = 4 (Im not quite sure, do i diff both then simultaneous or?)
This one actually has me a little stumped! If we look at the diagram below it doesn't really look like there is a common tangent (meaning, a point that is tangential to both the circle and the curve). If it does exist, your method is spot on - You would find an expression for the gradient of a tangent to the curve and the circle (the circle you would need to rearrange to make y the subject), and then put those equal to each other!
Edit: Or Rui has provided an alternative, but I don't think that it ends up being tangential to both, maybe?
2) Find the locus of P(x,y) and state any restrictions on a, given:
a) PA + PB= 1, where A(a,0) and B (-a,0)
b) ABS PA - PB=2, where A(a,0) and B (-a,0)
Do this by substituting the distance formula straight into the condition you are given (this gets really messy):
I'll let you take it from here - The algebra is disgusting but a HEAP of it should cancel out. You'll end up (I'm so sorry) with 16 terms on either side, but it should reduce to something fairly kind. But it is straight from the definition, and the other one will be similar!
Here's the equation in Wolfram if you need a hand with expanding it