Method 1:
Maybe the sketch reminds you of the previous activity? Let’s use Descartes’ advice: “Each problem that I solved became a rule which served afterwards to solve other problems.” So, if we can prove that P divides the ordinate in a constant ratio, we can use the result of Activity 14.

If the vertical from T cuts the X-axis in M, then:

In Activity 14 we proved that if in a circle with radius r, then the equation of the locus of P is:



This is the equation of an ellipse, so the locus is an ellipse!

Method 2:
It will here be useful to use parametric equations. Deduce the coordinates of P from the co-ordinates of S and T:

To find the Cartesian equation, i.e. to express y in terms of x, we need to eliminate q:
Square equations (1) and (2) (compare this process we used previously for the circle):