Question

Measure segments $F_{1}P$ and $F_{2}P$ and calculate the difference of their lengths. Drag the point P. What is special about the relationship between the lengths of these two segments for any hyperbola?

• A. The difference of the two segment lengths $F_{1}P$ and $F_{2}P$ is always the same.
• B. $|F_{1}P$ - $F_{2}P|$ is variable
• C. $|F_{1}P$ - $F_{2}P|$ is constant.
• D. $F_{1}P$ - $F_{2}P$ is constant.

Answer 1

Answer: (A), (C)

The correct options are
A The difference of the two segment lengths $F_{1}P$ and $F_{2}P$ is always the same.
C $|F_{1}P$ - $F_{2}P|$ is constant.

The difference of the two segment length, $F_{1}P$ and $F_{2}P$ is always the same. Therefore, the defining property of a hyperbola is the set of points $P$ for which $|F_{1}P-F_{2}P|$ is constant.