Definition:Hyperbola/Equidistance

Definition

 

Let $F_1$ and $F_2$ be two points in the plane.

Let $d$ be a length less than the distance between $F_1$ and $F_2$.

Let $K$ be the locus of points $P$ which are subject to the condition:

$\size {d_1 - d_2} = d$

where:

$d_1$ is the distance from $P$ to $F_1$

$d_2$ is the distance from $P$ to $F_2$

$\size {d_1 - d_2}$ denotes the absolute value of $d_1 - d_2$.

Then $K$ is a hyperbola.

The points $F_1$ and $F_2$ are the foci of $K$.

Also see

• Equidistance of Hyperbola equals Transverse Axis

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1 \text a$. Focal properties