Trace a moving midpoint locus from tangent intercepts

$dx/dt=2t$, $dy/dt=-2/t^2$, so $dy/dx=-1/t^3$. At the given point $t=p\ne0$. The tangent is $y-2/p=-(x-p^2)/p^3$, or $y=-x/p^3+3/p$.

At $y=0$, $x=3p^2$, so $Q=(3p^2,0)$. At $x=0$, $y=3/p$, so $R=(0,3/p)$.

The midpoint is $(3p^2/2,3/(2p))$. A locus is the set of all positions this midpoint occupies as the admissible p changes; it is not one fixed midpoint. Let its coordinates be x,y: $x=3p^2/2$, $y=3/(2p)$. Therefore $p=3/(2y)$ and $x=27/(8y^2)$, equivalently $y^2=27/(8x)$ with $x>0$, $y\ne0$. The teacher explicitly asks the lecturer to elaborate the meaning of “locus”; that interpretation is integrated here.