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Senior two mathematics ellipse exercises.
Since e=√3/2, we might as well set a = 2 and c = √ 3, then the elliptic equation becomes x? /4+y? = 1 ……①

This is a question of determining the chords of the score. Let the linear equation be x = √ 3+t and y = kt.

Combining with elliptic equation ①, and sorting, we get

(4k? + 1)t? +2√3t- 1=0 ……②

According to vector AF=3 and vector FB, there is tA=-3tB, which is obtained by combining with ②.

(-2√3-√△)/[2(4k? + 1)] = -3 (-2√3+√△)/[2(4k? + 1)]

Simplify with delta = 4 (2 √ 3)?

That is 12+4(4k? + 1)=4 12

Solve k? =2, because k>0, so k=√2.