(1) If the point m is on the right branch of the hyperbola,
As shown in the figure.
|MF 1| is the length of the line segment MF 1
|MF2| is the length of the line segment MF2,
The line segment MF 1 is longer than the line segment MF2.
∴|mf 1|>； |MF2|。
Therefore | MF1|-mf2 | > 0.
Remove the absolute sign, there are
| | MF 1 |-| MF2 | | = | MF 1 |-| MF2 |
When point m is on the right branch of hyperbola, | MF1|-| mf2 | = 2a (a >; 0)
(2) in the same way
|MF 1| is the length of the line segment MF 1
|MF2| is the length of the line segment MF2,
If point m is on the left branch of hyperbola, as shown below.
The line segment MF 1 is shorter than the line segment MF2.
∴|mf 1|<； |MF2|。
Therefore | MF1|-mf2 | < 0.
Remove the absolute sign, there are
| | MF 1 |-| MF2 | | =-(| MF 1 |-| MF2 |)=-2a。
When point m is on the left branch of hyperbola, | MF1|-mf2 | =-2a (a >; 0)
(3) The definition of hyperbola: || MF1|-mf2 | = 2a is different from the definition of ellipse: | mf 1 | | mf2 | = 2a has two aspects:
① The former is that the absolute value of the difference between any point M and two fixed points F 1 and F2 on the hyperbola is equal to a constant;
The latter is that the sum of the distances from any point m on an ellipse to two fixed points F 1 and F2 is equal to a constant,
One is the absolute value of the difference between two distances (that is, two line segments)
The other is the sum of two distances (that is, two line segments);
Totally irrelevant!
② The constant value 2a is different.
Hyperbola: 0
but
Ellipse: 0