This applet shows the graph of the parabola \(y=\dfrac{x^2}{4a}\) with its focus \(F\) at \((0,a)\) . The value of \(a\) can be changed by sliding the left slider.

The parabola can be rolled along the \(x\) -axis by moving the right slider.

As we roll the parabola in this way, what is the (cartesian equation of the) locus of the focus \(F\) ?

And as an even harder challenge, can you recreate this applet using GeoGebra or other graphing software?

To help you answer the first problem, we offer some questions to guide you. Attempt to answer each question before moving on to the next one.

Mini-question 1
Can you find the coordinates of a general point \(P\) on the parabola?

What do you think would be the most useful form to give the coordinates in?

Mini-question 2
Instead of the parabola rolling, how else could you think about the scenario?

Mini-question 3
What is the equation of the tangent through the point \(P\) ?

(If you found different forms of the point \(P\) in mini-question 1 which is most useful?)

What angle does it make with the \(x\) -axis?

Mini-question 4
What is the angle between \(PF\) and the tangent to the parabola at \(P\) ? (Is there a nice way to work this out?)

Mini-question 5
When the parabola has rolled so that the point \(P\) lies on the \(x\) -axis, what are the coordinates of the new location of \(P\) ?

Mini-question 6
Can you put all of your answers together to now work out the coordinates of the focus \(F\) when \(P\) lies on the \(x\) -axis?

This should give you enough to find the cartesian equation of the locus!