Finding the locus of a mid-point

Question

Let $A$ be the fixed point $(0, 4)$ and $B$ be a moving point $(2t, 0)$. Let $M$ be the mid-point of $AB$ and let the perpendicular bisector of $AB$ meet the $y$-axis at $R$. Find the locus of the mid-point $P$ of $MR$

My approach: I made some trial and error attempts. I feel that the locus is $x^2=-(y-2)$. But I am not sure. Also I wish to know how to actually solve the question.

Answer 1

1. Calculate the coordinates of $M$. $$M=(t,2).$$
2. Calculate the slope of line $AB$ ($m_{AB}$) and then of line $MR$ ($m_{MR}$). $$ m_{AB} = \frac{-4}{2t}=\frac{-2}{t}$$ $$ m_{RM} = \frac{t}{2}$$
3. Find out the equation of line $MR$ and then the $y$-intercept (point $R$). $$R=(0, \frac{4-t^2}{2}).$$
4. Find out point $P$. $$P=(\frac{t}{2}, \frac{8-t^2}{4})$$ And finally from the coordinates of $P$ express $y$ as a function of $x$.