Suppose that the center of the ellipse is at the origin o, the focus is on the X axis, the eccentricity is (√ 2) / 2, and the sum of the distances from one point P to two focuses on the ellipse is equal to √ 6. (1) if the straight line x + y + m = 0 intersects the ellipse at two points a and B, and OA ⊥ ob, find the value of M?

Suppose that the center of the ellipse is at the origin o, the focus is on the X axis, the eccentricity is (√ 2) / 2, and the sum of the distances from one point P to two focuses on the ellipse is equal to √ 6. (1) if the straight line x + y + m = 0 intersects the ellipse at two points a and B, and OA ⊥ ob, find the value of M?

From the known: 2A = √ 6, e = C / a = (√ 2) / 2, we can get a, B, and get the elliptic equation 4x ^ 2 / 6 + 4Y ^ 2 / 3 = 1
Let a (x1, Y1) B (X2, Y2)
From OA ⊥ ob: x1x2 + y1y2 = 0, that is, x1x2 + (x1 + m) (x2 + m) = 0... (1)
We can get x 1 + x 2 and x 1 x 2 by combining elliptic equation and linear equation, then we can get m by substituting them into (1)
(it's too hard to type these mathematical symbols. That's the way of thinking. Do it yourself.)