Known elliptical \ (C: \ dfrac {x ^ 2} {a ^ 2} + \ dfrac {y ^ 2} {b ^ 2} (a> b> 0) \) left and right respectively of focus \ (of F_1 , F_2, \)
through \ (of F_1 \) linear \ (L \) deposit \ (C \) in \ (a, B \) points. If \ (\ overrightarrow {AF_1} = \ dfrac {4} { 7} \ overrightarrow {AB}, | AF_2 | = | F_1F_2 |, \) is elliptical \ (C \) centrifugation was \ ((\ qquad) \)
\ (\ mathrm {a} \ {2} dfrac. {. 7} \ qquad \) \ (\ mathrm {B.} \ dfrac {. 3} {. 7} \ qquad \) \ (\ mathrm {C.} \ dfrac {. 4} {. 7} \) \ (\ mathrm { D.} \ dfrac {5} {7} \)
Analysis: a method of FIG
meaning of the questions known \ (| AF_2 | = 2c, \) in combination define an ellipse have
\ [| AF_1 | = 2 (
ac), \] and \ (\ overrightarrow {AF_1} = \ dfrac { . 4} {. 7} \ overrightarrow {AB} \) , then \ [| BF_1 | = \ dfrac {3} {2} (ac), | BF_2 | = \ dfrac {1} {2} (a + 3c). \]
take \ (AF_1 \) midpoint \ (E, \) connected \ (F_2E, \)
and because \ [| AF_2 | = | F_1F_2 |, \] Therefore $ \ mathrm {Rt} \ triangle EF_2A, \ mathrm {Rt} \ triangle EF_2B $ , applying the Pythagorean theorem to obtain
\ [AE ^ 2 + EF_2 ^
2 = AF_2 ^ 2, bE ^ 2 + EF_2 ^ 2 = BF_2 ^ 2, \] is substituted into available values and calculation \ (7c = 5a \ text {= C or A} \) $ ($ rounding \ (), \)
so \ (e = \ dfrac {5 } {7}, \) option \ (\ mathrm {D} \) is correct.
Method disposed two half ellipse focal length $ c $.
As shown, provided \ (\ angle AF_1F_2 = \ theta \) focal radius, elliptical equation by \ (\ mathrm {II} \ ) available
\ [| AF_1 | = \ dfrac {ep} {1
-e \ cos \ theta}, | BF1 | = \ dfrac {ep} {1 + e \ cos \ theta}, \] where \ (P \) is the in-focus elliptical pitch, \ (P = \ dfrac {B ^ 2} {C} \) . a \ (\ overrightarrow {AF_1} = \ dfrac {4} {7} \ overrightarrow {AB} \) available
\ [\ dfrac 43 = \ dfrac {| AF_1 |}. {
BF_1} = \ dfrac {1 + e \ cos \ theta} {1-e \ cos \ theta} \] Solutions have \ (e \ cos \ theta = \ dfrac 17 \) thereby. available
\ [2c = | F_1F_2 | = | AF_2 | = 2a- | AF_1 | = 2a- \ dfrac {ep} {1- \ frac 17} = 2a- \ dfrac 76 \ cdot \ dfrac {b ^ 2} { a}. \]
the \ (b ^ 2 = a ^ 2-c ^ 2 \) is substituted into and analyzed available \ ((. 5A-7C) (AC) = 0 \) , and therefore ask centrifugation was \ (\ dfrac57 \).