14
So the distance from the right focus to the line l : F 1 G= 2
2 2 − − B aC A b B a 2 2
2 2 2 2 ( ) ( ) B a A b +
The distance from the left focus to the line l : F 2 H= 2
2 2 − − − B aC A b B a 2 2
2 2 2 2 ( ) ( ) B a A b +
2 2 2 + + B aC A b B a 2 2
=
2 2 2 2 ( ) ( ) B a A b + Because point T(a , b) is on the ellipse.
2
2
2
a b
2 B B a A b A B b B a + = ⇒ = − 2 2 2 2 2
1 + = ⇒
2
2
2
2
A B
A
2 2 2 − − B aC A b B a 2 2
2 2 2 − − 2 2 ( ) ( ) B a A b FG F H B aC A b B a B aC A b B a + = = − − − + + 2 2 1 2 2 2 2 2 2 2 2 B aC A b B a
2 2
2 2
2
2 2 ( ) ( ) B a A b + 2 2
C
a A −
2 2 2 2 B aC A B A C B aC A B − = + 2 2
=
a A +
A
What’s more, an interesting thing can be found that the ratio of F 1 T and F 2 T is the same as the ratio above!
2
2
1 FT F T a C b − + = + + 2 2 ( ) a C b ( )
2
2
2
2
a Ca C b − + +
2 2
=
2
2
2
a Ca C b + + +
Because point T(a , b) is on the ellipse.
2
2
2
a b
2 B B a A b A B b B a + = ⇒ = − 2 2 2 2 2
1 + = ⇒
2
2
2
2
A B
A
So,
2
2 B a Ca C B a A B a Ca C B a A − + + − + + + − 2 2 2 2 2 2 2 2 2 2
2
1 2 FT F T
=
2
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