The eccentricity of the ellipse x 2A2 + y 2B2 = 1 (a > b > 0) is 32. The ellipse and the straight line x + 2Y + 8 = 0 intersect at P, Q, and | PQ | = 10. The equation of the ellipse is obtained

The eccentricity of the ellipse x 2A2 + y 2B2 = 1 (a > b > 0) is 32. The ellipse and the straight line x + 2Y + 8 = 0 intersect at P, Q, and | PQ | = 10. The equation of the ellipse is obtained

From C2 = A2-B2, A2 = 4B2. From x24b2 + y2b2 = 1x + 2Y + 8 = 0, eliminating x, 2Y2 + 8y + 16-b2 = 0. From the relationship between root and coefficient, Y1 + y2 = - 4, y1y2 = 16 − B22. | PQ | 2 = (x2-x1) 2 + (y2-y1) 2 = 5 (y1-y2) 2 = 5 [(Y1 + Y2) 2-4y1y2] = 10

The length of the minor axis of the ellipse ax ^ 2 + by ^ 2 = 1 is half of the length. The ellipse and the straight line x + 2Y + 8 = 0 intersect at points P, Q, and PQ = 10. The equation of the ellipse is obtained

From C / a = √ 3 / 2 = > (a ^ 2-B ^ 2) / A ^ 2 = 3 / 4
=>a^2=4*b^2
Then the elliptic equation can be written as x ^ 2 / (4 * B ^ 2) + y ^ 2 / b ^ 2 = 1
Let P (x1, Y1), q (X2, Y2)
From x ^ 2 / (4 * B ^ 2) + y ^ 2 / b ^ 2 = 1 and X + 2Y + 8 = 0, we can get the following results:
x^2-8x+32-2*b^2=0
According to Weida's theorem: X1 + x2 = 8, X1 * x2 = 32-2 * B ^ 2 --- (1)
The chord length formula is: (√ (1 + 1 / 4)) * √ ((x1 + x2) ^ 2-4 * X1 * x2) = √ 10 --- (2)
① The simultaneous solution is: B ^ 2 = 9
∴a^2=4*b^2=36
The equation of ellipse is: x ^ 2 / 36 + y ^ 2 / 9 = 1

P is the moving point on the circle C: x2 + y2 = 4, a (4.0), m satisfies the vector am = 2, the vector MP, find the trajectory equation of M
Is x squared plus y squared equal to 4

Let the coordinates of point p be (2cos θ, 2Sin θ). According to the meaning of the problem, vector AP = (2cos θ - 4,2sin θ) vector am = (4cos θ / 3-8 / 3,4sin θ / 3), so the coordinates of point m are (4cos θ / 3 + 4 / 3,4sin θ / 3). Let m (x, y), x = 4cos θ / 3 + 4 / 3, y = 4sin θ / 3, then (x-1) ^ 2 + y ^ 2 = (3 / 4) ^ 2

As shown in the figure, let p be a moving point on the circle x2 + y2 = 2, point d be the projection of P on the x-axis, and m be a point on the line PD, and PD = root 2 | MD | point a (0, root 2), F1 (- 1,0)
(1) There is a point F2 on the x-axis so that | MF1 | + MF2 | is the fixed value. Try to find the coordinate of F2 and point out the fixed value
(2) Find the maximum value of | Ma | + | MF1 |, and find out the coordinates of point m at this time

Let | ab | = 22 and the equation of the straight line AB be y-24-2 = x-13-1 &; X-Y 1 = 0. (3 points) in △ PAB, let the height of AB side be h, then 12 &; 22h = 10 &; H = 52, (7 points) let P (x, 0), then the distance from P to AB is | x 1 | 2, so | x 1 | 2 = 52, (10 points) the solution is x = 9, or x = - 11