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Find the trajectory equation of the midpoint m of a group of chords with slope 2, where the square of 8 / x plus the square of 4 / y of the ellipse equals 1
Let the coordinates of the intersection of the line with slope 2 and the ellipse be (x1, Y1) and (X2, Y2), and let the coordinates of the midpoint be (x, y)
Then X1 ^ 2 / 8 + Y1 ^ 2 / 4 = 1, X2 ^ 2 / 8 + Y2 ^ 2 / 4 = 1
The difference between the two formulas is (x1 + x2) (x1-x2) / 8 = - (Y1 + Y2) (y1-y2) / 4
And (y1-y2) / (x1-x2) = 2x1 + x2 = 2x Y1 + y2 = 2Y
So x + 4Y = 0 is the trajectory equation of M
Given the ellipse x2 / 8 + Y2 / 4 = 1, find the trajectory equation of the midpoint of the chord with slope 2
Let the chord of the ellipse X & # 178 / 8 + Y & # 178 / 4 = 1 be AB, and the coordinates of a and B are (x1, Y1) (X2, Y2) respectively
Because the slope of AB is 2, let the equation of the line where AB is located be y = 2x + B
Substituting it into the elliptic equation, we get
x²+2(2x+b)²=8
x²+8x²+8bx+2b²=8
9x²+8bx+2b²-8=0
Since a and B are on the ellipse, X1 and X2 are the two roots of the equation
x1+x2= -8b/9
y1+y2=2(x1+x2)+2b =-16b/9+2b=2b/9
Let the midpoint of AB be p (x, y)
be
x=(x1+x2)2
y=（y1+y2)/2
Then Y / x = (2B / 9) / (- 8B / 9) = - 1 / 4
Therefore, the trajectory equation of the midpoint AB is
y=-1/4 x
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