12.11 Distance to this point minus distance to that point is

On the drawing below are the points (5,0) and (–5,0). We’re going to draw yet another shape—our final conic section. In order to be on our shape, the point ( $x$ , $y$ ) must have the following property:

Take the distance from ( $x$ , $y$ ) to (5,0), and the distance from ( $x$ , $y$ ) to (–5,0). Those two distances must differ by 6. (In other words, this distance minus that distance must equal ±6.)

We’re going to start this one the same way we did our other shapes: intuitively. Your object is to find all the points that have that particular property. Start by finding the two points on the $x$ -axis that work. After that, you have to sort of dope it out as we did before.

When you think you know the shape, don’t call it out! Call me over and I will tell you if it’s right.