If a straight line passing through the left focus of the ellipse x ^ 2 + 2Y ^ 2 = 4 with a left inclination angle of 30 degrees intersects the ellipse at two points a and B, the chord length AB will be obtained=

If a straight line passing through the left focus of the ellipse x ^ 2 + 2Y ^ 2 = 4 with a left inclination angle of 30 degrees intersects the ellipse at two points a and B, the chord length AB will be obtained=

The standard equation is: X & # 178 / 4 + Y & # 178 / 2 = 1
c²=a²-b²=2
So, the left focus F1 (- √ 2,0)
So, AB: y = (√ 3 / 3) (x + √ 2)
Let a (x1, Y1), B (X2, Y2)
AB²=(x1-x2)²+(y1-y2)²
Because a and B are on the straight line y = (√ 3 / 3) (x + √ 2), so: Y1 = (√ 3 / 3) (x1 + √ 2), y2 = (√ 3 / 3) (x2 + √ 2)
Then: y1-y2 = (√ 3 / 3) (x1-x2)
Therefore, AB & # 178; = (4 / 3) (x1-x2) &# 178;
y=(√3/3)(x+√2)
x²/4+y²/2=1
By eliminating y, we get: 5x & # 178 / 6 + (2 √ 2) x / 3-4 / 3 = 0
x1+x2=-(4√2)/5,x1x2=-8/5
Then: (x1-x2) &# 178; = (x1 + x2) &# 178; - 4x1x2 = 64 / 25
So AB & # 178; = (4 / 3) (x1-x2) &# 178; = 256 / 75
Therefore, chord length AB = (16 √ 3) / 15