The trajectory equation of the midpoint of a group of parallel strings with slope k of ellipse x2 + Y2 / 4 = 1 is a part of the straight line 2x + y = O, and K is obtained

The trajectory equation of the midpoint of a group of parallel strings with slope k of ellipse x2 + Y2 / 4 = 1 is a part of the straight line 2x + y = O, and K is obtained

Let Y / 2 = y '
Y = 2Y ', to establish a new coordinate
Ellipse becomes circle x ^ 2 + y '^ 2 = 1
The trajectory equation of the midpoint of a parallel string is a straight line x + y '= o
Because the line x + y '= O passes through the center of the circle, the parallel string is perpendicular to the line. The equation of parallel string is y' = x + B
That is, in the original coordinates: y = 2x + 2B
k=2

In the ellipse x2 + 2Y2 = 1, the trajectory equation of the midpoint of the parallel chord with slope 2 is

Let the slope 2 equation y = 2x + K
Substituting into ellipse, x ^ 2 + 2 (2x + k) ^ 2 = 1
9x^2+8kx+2k^2-1=0
Sum of equations X1 + x2 = - 8K / 9
Then the abscissa of the midpoint (x1 + x2) / 2 = - 4K / 9
If the midpoint is y = 2x + K, then the midpoint is longitudinally K / 9
Let (x, y) be the midpoint,
x= - 4k/9,y= k/9
Midpoint equation x + 4Y = 0

It is known that the elliptic equation is x2 / 4 + y2 = 1, and the trajectory equation of the midpoint of a parallel string with a slope of root 3 is obtained

Let y = √ 3x + B be the chord of the ellipse, where the coordinates of the points are (m, n). The simultaneous result is y = √ 3x + BX & # 178; = 4 + Y & # 178; = 1, and the elimination of Y results in 13X & # 178; + 8 √ 3bx + 4B & # 178; - 1 = 0, so 2m = - 8b √ 3 / 13, M = - 4B √ 3 / 13N = √ 3 (- 4B √ 3 / 13) + B = B / 13, so B = 13N, so m = (- 4 √ 3 / 13) * 13N = -

Find the trajectory equation of the midpoint of the chord cut by the ellipse x2 + y2 = 2

Let the coordinates of the two ends of the chord be (x1, Y1), (x2. Y2), and the coordinates of the middle point of the chord be (x, y). The slope of the straight line where the chord is located is kx1 & sup2; + Y1 & sup2; = 2x2 & sup2; + Y2 & sup2; = 2. Subtracting (x1 + x2) (x1-x2) + (Y1 + Y2) (y1-y2) = 0y1-y2 / x1-x2 = - (x1 + x2) / (Y1 + Y2) k = Y-2 / x, and K = - (x1 + x2) / (Y1 + Y2) X1 +