A-level Maths Question

Hi, can someone please explain part (b) to me? I can't seem to get my head around it. Thanks. I will attach the graphs I plotted to try and help me.

For any point, x^2+y^2 is the (squared) distance from the origin, so simply sub y^2 from your parametric equation into that and maximize with respect to x. The distace from the origin is the radius of the corresponding circle.

Ok…when you say maximise with respect to x, would that be the maximum value of dr/dx ?

x^2 + y^2 is the squared distance from O which you can call r^2 if you want to. Differentiating it with respect to x (after subbing for y^2) gives the maximum (stationary point) value as using the chain rule
d r^2 / dx = 2 r dr/dx
so it equals zero if r or dr/dx is zero.

Ok I think that is starting to make more sense now. But isn’t the roots of the difference between two functions the same x value as the intersection? So on the graph, when is r^2 maximum? Or is trying to see it on a graph too confusing?

You could do the question in two ways, using calculus so maximize r^2 where r refers to the distance from the origin (it obviously also corresponds to a circle radius). So as per the previous posts / model solution.

Or when a circle touches the parabola so when
-9/4 x^4 +10x^2 = r^2
Doing b^2-4ac = 0 gives the same result as the calculus based but is probably simpler. I guess this is what youre referring to when youre talking about the difference beween two functions/intersection/...? If not, youll have to be more specific.

Yep that’s what I meant. Thanks that makes more sense 👍🏼

As its a hidden quadratic, the noncalculus quadratic/touching is indeed simpler. Though the calculus based approach is more genera (not just for (hidden) quadratics)l, so its worth making sure you understand it.

Hi, sorry, thanks for your help so far but I still don't feel like this is clicking fully. So, after you let each y^2 equation equal each other, you have r^2=10x^2-(9/4)x^2. This tells you how r changes with x. Here r is, as you previously said, the squared distance from the centre (0,0), but isn't the radius constant? No matter where on the circle, the distance from the centre to that point is the same? When letting the parametric equation equal the circle by letting the y^2 terms equal each other, I am confused what this new equation is telling us? Normally you let equations equal to find a point they intersect, which I do understand, but I still don't understand the rest.

Wait, is r maximum because if it was any larger, the parametric graph and circle wouldn’t intersect, and if we solved for a smaller value, this would just give another r value for a smaller circle which intersects the parametric graph 8 times?