The linear equation with ellipse (x ^ 2 / 9) + (y ^ 2 / 4) = 1 intersecting at two points a and B, and the midpoint of line AB is m (1,1) is, using the point difference method

The linear equation with ellipse (x ^ 2 / 9) + (y ^ 2 / 4) = 1 intersecting at two points a and B, and the midpoint of line AB is m (1,1) is, using the point difference method

Point difference method
Let a x1, Y1, B X2, Y2 be substituted into two equations to reduce X1 + x2 = 2, Y1 + y2 = 2
It is proved that the slope k = - 9 / 4 and passing through (1,1) points
Another method:
Let Y-1 = K (x-1)
y=kx+(1-k)
Bring in the ellipse 4x & # 178; + 9y & # 178; = 36
(4+9k²)x²+18k(1-k)x+9(1-k)²-36=0
x1+x2=-18k(1-k)/(4+9k²)
Abscissa of midpoint = (x1 + x2) / 2
So - 9K (1-k) / (4 + 9K & # 178;) = 1
-9k+9k²=4+9k²
k=-4/9
So 4x + 9y-13 = 0

Let a and B be fixed points on the ellipse x ^ 2 / 4 + y ^ 2 = 1, and point m (1,1 / 2) be the middle point of line ab. then the linear equation of AB is obtained
Please write down the process of solving the problem

Let a (x1, Y1), B (X2, Y2)
(x1) & sup2; (4 + (Y1) & sup2; = 1, (x2) & sup2; (4 + (Y2) & sup2; = 1
(x1-x2)(x1+x2)/4+(y1-y2)(y1+y2)=0.
∵ X1 + x2 = 2, Y1 + y2 = 1, let (y1-y2) / (x1-x2) = k,
∴(2/4)+k=0,
∴k=-1/2.
According to the point oblique equation, x + 2y-2 = 0

It is known that the elliptic equation is MX & # 178; + NY & # 178; = 1, the line y = x + 1 intersects the ellipse at two points a and B, and the abscissa of the midpoint m of the line AB is - 3 / 4,
And satisfy OA vertical ob, find the equation of the ellipse!

L:y=x+1
A(a,a+1),B(b,b+1)
xM=(a+b)/2=-3/4
a+b=-1.5
b=-1.5-a
[(a+1)/a]*[(b+1)/b]=-1
2ab+a+b+1=0
2ab-1.5+1=0
2ab-0.5=0,a