As shown in the figure, we know that m (m, m ^ 2), n (n, n ^ 2) are two different points on the parabola C: y = x ^ 2, and m ^ 2 + n ^ 2 = 1, M + n ≠ 0, l is the vertical bisector of Mn Let the equation of ellipse e be x ^ 2 / 2 + y ^ 2 / a = 1 (a > 0, a ≠ 2) 1. When m and N move on the parabola C, find the value range of the slope k of the straight line L 2. It is known that line L and parabola C intersect at two different points a and B, and l and ellipse e intersect at two different points P and Q. let the midpoint of AB be r, and the midpoint of PQ be s. if vector or · vector OS = 0, the range of eccentricity of E can be calculated

As shown in the figure, we know that m (m, m ^ 2), n (n, n ^ 2) are two different points on the parabola C: y = x ^ 2, and m ^ 2 + n ^ 2 = 1, M + n ≠ 0, l is the vertical bisector of Mn Let the equation of ellipse e be x ^ 2 / 2 + y ^ 2 / a = 1 (a > 0, a ≠ 2) 1. When m and N move on the parabola C, find the value range of the slope k of the straight line L 2. It is known that line L and parabola C intersect at two different points a and B, and l and ellipse e intersect at two different points P and Q. let the midpoint of AB be r, and the midpoint of PQ be s. if vector or · vector OS = 0, the range of eccentricity of E can be calculated

1. The slope of the line Mn is m + n ≠ 0, so the slope k of the line L is - 1 / (M + n). From the inequality (M + n) ^ 2 ≤ 2 (m ^ 2 + n ^ 2) = 2, we know - √ 2 ≤ M + n ≤ √ 2. So the range of the slope k of L is k ≤ - √ 2 / 2 or K ≥ √ 2 / 2
2. We also know that the midpoint of M and N is ((M + n) / 2,1 / 2), so the equation of linear l can be written as x + (M + n) y - (M + n) = 0
If it is combined with parabola C, we get x ^ 2 + X / (M + n) - 1 = 0, so X1 + x2 = - 1 / (M + n), x1x2 = - 1, Y1 + y2 = X1 ^ 2 + x2 ^ 2 = (x1 + x2) ^ 2-2x1x2 = 2 + 1 / (M + n) ^ 2, so the coordinates of R are (- 1 / [2 (M + n)], 1 + 1 / [2 (M + n) ^ 2])
In conjunction with the ellipse e, we obtain [a + 2 / (M + n) ^ 2] x ^ 2-4 / (M + n) + 2-2a = 0 and [a (M + n) ^ 2 + 2] y ^ 2 - [2A (M + n) ^ 2] y + a (M + n) ^ 2-2a = 0, so X3 + X4 = 4 (M + n) / [a (M + n) ^ 2 + 2], Y3 + Y4 = 2A (M + n) ^ 2 / [a (M + n) ^ 2 + 2], so the coordinates of s are (2 (M + n) / [a (M + n) ^ 2 + 2, a (M + n) ^ 2 / [a (M + n) ^ 2 + 2])
According to the relation or · OS = 0, the range of a = 2 / [2 (M + n) ^ 2 + 1] is [2 / 5,2]
Then the square of eccentricity e of E is 2 / (2 + a), and the value range is (1 / 2,5 / 6]
Finally, the range of eccentricity e of E is (√ 2 / 2, √ 30 / 6]

It is known that the intersection of parabola y = -- (X -- m) ^ 2 + 1 and X axis is a, B (B is on the right side of a), and the intersection of parabola y axis is C? When point B is on the right side of the origin and point C is below the origin, is there a triangle BOC, which is an isosceles triangle? If it exists, find out the value of M; if it does not exist, please explain the reason

Solution: from the intersection of the parabola y = - (X -- m) ^ 2 + 1 and the X axis as a, B, we can get the following result:
(x-m)^2=1
x-m=±1
x=m±1
And B is on the right side of a, so the coordinates of a and B are a (m-1,0) and B (M + 1,0) respectively
The intersection of the parabola and the y-axis is C,
y=-(0-m)^2+1=1-m^2
So the coordinates of point C (0,1-m ^ 2)
According to the coordinates of the three points, a (m-1,0), B (M + 1,0), C (0,1-m ^ 2)
According to the distance formula between two points
AB^2=|m+1-(m-1)|^2=2^2=4
AC^2=(m-1)^2+(1-m^2)^2
BC^2=(m+1)^2+(1-m^2)^2
(1) When AC = BC
That is, AC ^ 2 = BC ^ 2
(m-1)^2+(1-m^2)^2=(m+1)^2+(1-m^2)^2
(m-1)^2=(m+1)^2
4m=0
m=0
Then the coordinates of a, B and C are a (- 1,0), B (1,0) and C (0,1) respectively
It is known that C is below the origin, so m = 0 does not conform to the meaning of the problem
(2) When AB = AC
That is: ab ^ 2 = AC ^ 2
(m-1)^2+(1-m^2)^2=4
m^2-2m+1+1-2m^2+m^4-4=0
m^4-m^2-2m-2=0
m^2(m^2-1)-2(m+1)=0
m^2(m+1)(m-1)-2(m+1)=0
(m+1)(m^2(m-1)-2)=0
So: M = - 1, or m ^ 2 (m-1) - 2 = 0,
And: m ^ 2 (m-1) - 2 = 0
m^3-m^2-2=0
Just work out the value of M
Point B is on the right side of the origin, and point C is below the origin, that is: M + 1 > 0, M > - 1; 1-m ^ 21
When AB = BC
That is ab ^ 2 = BC ^ 2
4=(m+1)^2+(1-m^2)^2
4=m^2+2m+1+1-2m^2+m^4
m^4-m^2+2m-2=0
m^2(m^2-1)+2(m-1)=0
m^2(m+1)(m-1)+2(m-1)=0
(m-1)(m^2(m+1)+2)=0
M = 1 or m ^ 2 (M + 1) + 2 = 0
M = 1 does not meet the condition,
When m > 1, m ^ 2 (M + 1) + 2 has no solution

Given that circle C passes through fixed point a (0, a) and the length of chord Mn cut on x-axis is 2a, if am = m, an = n, M / N + n / M is the largest, the equation of circle C is

Let the center of circle C be (x, y) and the radius be r
∵ circle C passes through point a (0, a), ∵ (0-x) 2 + (a-y) 2 = R2
The length of the chord Mn cut by the circle C on the x-axis is 2A
The point (x + A, 0) is on the circle C, that is, (x + A-X) 2 + (0-B) 2 = R2
Then there is x2 + (a-y) 2 = A2 + Y2, that is, X2 = 2ay
The trajectory equation of the center C of circle C is x2 = 2ay