It is known that the eccentricity of ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) e = √ 6 / 3, and the distance between the line passing through points a (0, b) and B (a, 0) and the origin is √ 3 / 2 Given the fixed point E (- 1,0), if the line y = KX + 2 (K ≠ 0) intersects the ellipse at two points c and D, ask: is there a value of K to make the circle with diameter CD pass through point e? Please explain the reason

It is known that the eccentricity of ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) e = √ 6 / 3, and the distance between the line passing through points a (0, b) and B (a, 0) and the origin is √ 3 / 2 Given the fixed point E (- 1,0), if the line y = KX + 2 (K ≠ 0) intersects the ellipse at two points c and D, ask: is there a value of K to make the circle with diameter CD pass through point e? Please explain the reason

Let the coordinates of CD be (x1, Y1), (X2, Y2) EC = (x1 + 1, Y1), ed = (x2 + 1, Y2), EC and ED be vectors if e is on the circumference of a circle with diameter CD, We have EC * ed = 0 (x1 + 1) (x2 + 1 + 1) (x + 1 + 1) + y1y2 = 0 (x1 + 1 + 1) (x2 + 1) (x2 + 1) + Y1, 2 = 0 (x1 + 1 + 1) (x2 + 1) (x2 + 1) + 1Y2 = 0 (x1 + 1 + 1) (x2 + 1) + 1 + y1y2 = 0 (x1 + 1) (x1 + 1) (x2 + 1) + 1 + y1y2 = 0, x + 2 = 0, X1 + 2 = 0, and 1 + 1 (1 / 3 + 3 + 3 + 3 + 3 + 1 (1 / 3 + 3 + 3 + 3 + 1 (1 / 3 + 3 + 3 + 3 + 1 (1 / 3 + 3 + 3 + k) x + 1 (1 (1 / 3 + 3 + 3 + 3 + 3 + 3 + 1 + 3 + 3 + 3 + 1 + 3 + 1 + 3 + 3 + 1 + 3 + 1 + 3 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + = 0 (14 / 3) - 4K = 0, k = 7 / 6