Recently I was going through co-ordinate geometry. While studying about parabola I encountered something called "parameteric coordinates". In my textbook some parameteric coordinates were given [y²=4ax : (at², 2at)] for specific equations of parabola, but nothing related to how these coordinates are defined was there.
Searched a lot but cant find anything convincing, one of many posts said it's done by defining x and y in terms of an other variable where x(t)=t itself and y(t)=a*x(t)² => y=at² from standard eqn of parabola.
Well, it does not make any sense as (t,at²) isn't any parameteric coordinate.
I am really confused (to the extent where I can no longer differentiate between maths and biology) Please someone help me.
I need an elaborate answer ASAP. (I am no mathemagician please be clear and don't use heavy terminology)
Thanks
In the example you gave for a parabola, $t$ is the parameter. It looks like you have the right understanding of how parametric coordinates work (ie each point on a curve is given by two functions of a parameter; in your parabola example, if you put in, say, $t=3$, you get the point $(9a,6a)$). However, I don't think a parabola is the best example of why these can be helpful, even among conics.
Let's start with a circle with radius $r$ centred at $(0,0)$. The Cartesian equation for this is $x^2+y^2=r^2$. But this is an implicit formula; you have to do some calculation to get $y$ in terms of $x$. OK; rearrange and we get $$y=\sqrt{r^2-x^2}$$ ...and plotting this, we get a semicircular arc. In fact, for every $-r<x<r$, we need two corresponding $y$ values; we can write this as $$y=\pm\sqrt{r^2-x^2}$$ but it doesn't change the fact that we can't write $y$ as a function of $x$ (technically, a function needs to just have one output for a given input).
Here's the parametric form for the same circle: $$x=r\cos{t},\quad y=r\sin{t}$$
Hopefully you can see why this works. Now, for every value of $t$, we get exactly one point on the circle, which makes this form far easier to work with (in certain applications - eg physics of circular motion; in fact, one of the common uses for parametric coordinates is in modelling motion, where the parameter is often time).
Now, in this case, $t$ corresponds exactly to the angle the line joining the point on the circle to $O$ makes with the positive $x$-axis; but this isn't a necessary property of the parameterisation. We could just as well say $$x=r\cos{2t+5},\quad y=r\sin{2t+5}$$ and get the same circle.
Similarly, it's easy to parameterise an ellipse: $$x=a\cos{t},\quad y=b\sin{t}$$
(just scaling the circle) or a hyperbola: $$x=a\cosh{t},\quad y=b\sinh{t}$$
The similarities between these forms are another reason it can be useful to use parametric coordinates for conics (and other curves).
In case you're not familiar with $\cosh$ and $\sinh$, another form for the hyperbola is $$x=a\sec{t},\quad y=b\tan{t}$$
Another advantage of parametric coordinates is it is relatively easy to shift and rotate curves by using matrix transformations (not every ellipse has axes parallel to the coordinate axes, after all).
I hope that's useful - let me know if anything isn't clear!