Finding the locus of a mid-point 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.

*

*Calculate the coordinates of $M$.
$$M=(t,2).$$

*Calculate the slope of line $AB$ ($m_{AB}$) and then of line $MR$ ($m_{MR}$).
$$ m_{AB} = \frac{-4}{2t}=\frac{-2}{t}$$
$$ m_{RM} = \frac{t}{2}$$

*Find out the equation of line $MR$ and then the $y$-intercept (point $R$).
$$R=(0, \frac{4-t^2}{2}).$$

*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 $x$.

