It is known that the line L: y = x, the ellipse C center is at the origin, the focus is on the x-axis, the focal length is 2, and the short axis length is 2 √ 3 (1) The left focus of ellipse C is F1, the right vertex is B, and the parallel line L 'of straight line L passing through point F1. If l' intersects ellipse C with m and N, calculate the area of triangle MNB

It is known that the line L: y = x, the ellipse C center is at the origin, the focus is on the x-axis, the focal length is 2, and the short axis length is 2 √ 3 (1) The left focus of ellipse C is F1, the right vertex is B, and the parallel line L 'of straight line L passing through point F1. If l' intersects ellipse C with m and N, calculate the area of triangle MNB

From the meaning of the title, we know that 2C = 2, 2b = 2 √ 3, so B ^ 2 = 3, a ^ 2 = 4
So the standard equation of ellipse is: x ^ 2 / 4 + y ^ 2 / 3 = 1
So its left focus F1 (- 1,0), right vertex B (2,0)
The line L 'passing through point F1 is parallel to y = x, so the equation of L' is y = x + 1
Substituting y = x + 1 into x ^ 2 / 4 + y ^ 2 / 3 = 1, 7x ^ 2 + 8x-8 = 0
x1+x2=-8/7,(x1)(x2)=-8/7
Further, we obtain that Y1 + y2 = (x1 + 1) + (x2 + 1) = 6 / 7, (Y1) (Y2) = (x1 + 1) + (x2 + 1) = (x1) (x2) + (x1 + x2) + 1 = - 9 / 7
So | Mn | = root [(x2-x1) ^ 2 + (y2-y1) ^ 2] = root {[(x2 + x1) ^ 2-4 (x2) (x1)] + [(Y2 + Y1) ^ 2-4 (Y2) (Y1)]}
=24/7
The distance from the vertex B to the line y = x + 1 is the height of the bottom Mn, which is (3 / 2) times the root 2, so the area of the triangle MNB is
S = {(24 / 7) * [(3 / 2) times root 2]} / 2 = (18 / 7) times root 2