The line L passing through the left focus F of the ellipse x ^ 2 / 9 + y ^ 2 = 1 intersects the ellipse at P and Q. if the distance between P and Q is equal to the length of the minor axis of the ellipse, the inclination angle of the line L is calculated Please attach the solution,

The line L passing through the left focus F of the ellipse x ^ 2 / 9 + y ^ 2 = 1 intersects the ellipse at P and Q. if the distance between P and Q is equal to the length of the minor axis of the ellipse, the inclination angle of the line L is calculated Please attach the solution,

Long axis a = 3, short axis B = 1, C = 2 √ 2, left focus F1 (- 2 √ 2,0), | PQ | = 2,
Let the slope of PQ equation K, y = K (x + 2 √ 2), and substitute it into elliptic equation, x ^ 2 + K ^ 2 (x + 2 √ 2) ^ 2 = 1,
(9k^2+1)x^2+36√2k^2x+72k^2-9=0,
Let P (x1, Y1), q (X2, Y2), then x1, X2 are two roots of quadratic equation,
According to Weida's theorem, X1 + x2 = - 36 √ 2K ^ 2 / (9K ^ 2 + 1), X1 * x2 = (72K ^ 2-9) / (9K ^ 2 + 1),
According to chord length formula | PQ | = √ (1 + K ^ 2) (x1-x2) ^ 2
=√(1+k^2)[(x1+x2)^2-4x1x2]
=√(1+k^2){[(-36√2k^2/(9k^2+1)]^2-4(72k^2-9)/(9k^2+1)}
=[√(1+k^2)(36K^2+36)]/(9k^2+1)
2(9k^2+1)=6(1+k^2)
k=±√3/3,
tanα=±√3/3,
α = 30 ° or α = 150 °,
The inclination of line L is 30 or 150 degrees

For known points P (4, - 9) and Q (- 2,3), then the ratio of the intersection m of the Y-axis and the linear segment PQ to the linear segment PQ is?

Let the line passing through two points be (Y-3) / (- 9-3) = (x + 2) / (4 + 2), and simplify to y = - 2x-1, so the coordinate of point m is (0, - 1) | PM | = √ [(4) ^ 2 + (- 9 + 1) ^ 2] = 4 √ 5 | QM | = √ [(- 2) ^ 2 + (3-1) ^ 2] = 2 √ 2 | PM | / | QM | = (4 √ 5) / (2 √ 2) = √ 10m. The ratio of line segment PQ is √ 10:1