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Let u= $\langle X,Y \rangle$ and v= $\langle X_1,Y_1 \rangle$. Describe the set of points $(X,Y)$ in 2-space that satisfy the stated conditions:

$(a)$ ||u - v||$=1$

$(b)$ ||u - v||$≤1$

$(c)$ ||u - v||>1

I don't know how to answer these questions. I see that the answer to $(a)$ would be two concentric circles where the difference between their respective radii would be 1, but I don't know how to answer the question. I am even more lost on $(b)$ and $(c)$. Any help would be appreciated. Thank you.

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    I assume $X_1$, $Y_1$ are fixed numbers. Then (a) is tthe circle with center $(X_1,Y_1)$, radius $1$. (b) is same circle plus its interior. (c) is all points outside the same circle. Can't say much more, hard to type in TeX without feedback.2011-07-06

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The question is not entirely clear. I will take it to mean, $\bf v$ is given, describe the set of $\bf u$ such that $\|{\bf u}-{\bf v}\|=1$, etc. If my interpretation is incorrect, perhaps OP will clarify and we can make some progress.

So anyway the set of $\bf u$ such that $\|{\bf u}-{\bf v}\|=1$ is a circle of radius 1, centered at $\bf v$. In terms of the $x$, $y$ variables, it's the set of $(x,y)$ such that $(x-x_1)^2+(y-y_1)^2=1$.

Now can you see what to do with the other two parts of the question?

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    I don't blame you for the unclear wording. Do you understand why the set of $\bf u$ such that $\|{\bf u}-{\bf v}\|=1$ is a circle centered at $\bf v$? Do you understand why that leads to the equation $(x-x_1)^2+(y-y_1)^2=1$? Did the comment from user6312 do anything for you?2011-07-07