Curve of possible locations of a post office between two buildings

Question

One of my students had to answer this question, and it is actually stumping me.

Let $A,B$ be two buildings 10 miles apart. Suppose $P$ is a post office such that the distance between $A$ and $P$ is 2 miles more than the distance between $P$ and $B$. Then what shape is the curve of possible locations of $P$?

My instinct is that it should be an ellipse or (perhaps) a hyperbola, but I cannot see this formally.

Answer 1

Your desired locus is one branch of a hyperbola.

You can see this by placing A and B at convenient points on the Cartesian plane--say A at (-5, 0) and B at (5, 0). Place P at (x, y). Use the distance formula to get an equation for your locus:

$$\sqrt{(x--5)^2+(y-0)^2} = 2 + \sqrt{(x-5)^2+(y-0)^2}$$

Simplify that equation by the usual method of squaring both sides of the equation, simplifying, isolating the square root, squaring both sides, and simplifying. You will get an equation of a hyperbola:

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

You then note that, in the original equation (or using geometric considerations), $x$ must be positive. This limits the locus to the right branch of the hyperbola. With some more work you can show that each point on the right branch satisfies the original equation, so the locus is the entirety of the right branch.

I'll leave the details to you. Ask if you need more help.