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let $A$ be the point $(0, 4)$ in the $xy$-plane and let $B$ be the point $(2t, 0)$. Let $L$ be the midpoint of $AB$ and let the perpendicular bisector of $AB$ meets the $y$-axis at $M$. Let $N$ be the midpoint of $LM$. Find the locus of $N$ (as $t$ varies).

I was trying this question many times. I don't know where I have to start. I was thinking that this the locus of circle , because it moving around the midpoint $LM$. I don't know how to do this question.. If anybody help me I would very thankful to him.

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We obtain: $$L\left(\frac{0+2t}{2},\frac{4+0}{2}\right)$$ or $$L(t,2)$$ and since $$m_{AB}=\frac{4-0}{0-2t}$$ or $$m_{AB}=-\frac{2}{t},$$ we obtain $m_{ML}=\frac{t}{2}$ and the equation of the line $ML$ is $$y-2=\frac{t}{2}(x-t),$$ which for $x=0$ gives $y_{M}=2-\frac{t^2}{2}$.

Thus, $M(0,2-\frac{t^2}{2})$, $N\left(\frac{t}{2},2-\frac{t^2}{4}\right)$, which gives the answer: $$y=2-x^2.$$

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