Given the ellipse X & # 178; + 2Y & # 178; = 4, then a point (1,1) in the ellipse is the length of the chord of the midpoint?

Given the ellipse X & # 178; + 2Y & # 178; = 4, then a point (1,1) in the ellipse is the length of the chord of the midpoint?

Chord AB midpoint (1,1) a (x1, Y1) B (X2, Y2)
AB linear equation: Y-1 = K (x-1)
y=k(x-1)+1 y1-y2=k(x1-x2)
x^2+2k^2(x-1)^2+2k(x-1)+1-4=0
(1+2k^2)x^2 -(4k^2-2k)x+2k^2-2k-3=0
x1+x2=(4k^2-2k)/(1+2k^2)
x1x2=(2k^2-2k-3)/(1+2k^2)
(x1+x2)/2=1 x1+x2=2
(4k^2-2k)/(1+2k^2)=2
-2k=2
k=-1
x1x2=1/3
(x1-x2)^2=(x1+x2)^2-4x1x2=2^2-4*(1/3)=4-4/3=8/3
|AB|^2=(k^2+1)(x1-x2)^2=2*8/3=16/3
|AB|=4/√3

The coordinates of the midpoint of the line segment cut by the ellipse x2 + 2Y2 = 4 are______ ．

By substituting the line y = x + 1 into the ellipse x2 + 2Y2 = 4, we can get 3x2 + 4x-2 = 0, △ = 16 + 24 > 0  X1 + x2 = - 43, and the abscissa of the midpoint is: − 23. By substituting the line equation, we can get the ordinate of the midpoint is − 23 + 1 = 13, and the midpoint coordinate of the line segment cut by the ellipse x2 + 2Y2 = 4 is (− 23, 13). So the answer is: (− 23, 13)

The center of the ellipse is at the origin, the focus is on the x-axis, the ellipse section line C: x + 2y-2 = 0, the chord length is 5, and the chord midpoint coordinates (1,1 / 2) are used to solve the elliptic equation
Writing process