Simply speaking, the whole idea behind parametrics is the ability to express the (x,y) coordinates in terms of an alternative coordinate , which for the matter of HSC, a single parameter, say t.
What this means is that...ermmm.. take this situation. Say, a rich man like Bill Gates own houses across the globe. He chooses to live in each house according to that day's temperature t. And, we use (x,y) to define the longitude and latitude of the location of his residence. Hence, we can say his resort is located at (x,y) which is a function of t. i.e. (x,y) = (f(t),g(t)) or x = f(t) & y = g(t).
Questions in the hsc, will often give you a number of conditions, tangents crossing and normals crossing etc.. THe aim is to express the intersection point (or whatever the condition is) in terms of the parameters (p,q if two tangents exist). and then using algebraic simplification, simulataneous equations, roots of quadratics, discriminants to find the equation(s) which define the point of intersection traces as it moves.
As previous replies , x = 2ap & y = ap^2 defines a point which traces out the parabola x^2 = 4ay.
Another example is x = acost & y = bsint defines an ellipse of equation x^2/a^2 + y^2/b^2 =1.
2005 HSC Graduate (Killara)
EngAdv 82 MX1 98 MX2 95 Phys 93 Chem 90 FrCont 67
B.E. (Biomedical) B.Sc. (Advanced Mathematics) USYD