It is known that a straight line y = X-1 and an ellipse (x ^ 2 / M) + (y ^ 2 / m-1) intersect at two points a and B. If a circle with diameter AB passes through the focus F of the ellipse, then the value of M is

It is known that a straight line y = X-1 and an ellipse (x ^ 2 / M) + (y ^ 2 / m-1) intersect at two points a and B. If a circle with diameter AB passes through the focus F of the ellipse, then the value of M is

Standard practice
(x ^ 2 / M) + (y ^ 2 / m-1) = 1 to
（2m-1)x^2-2mx+2m-m^2=0
Weida theorem: x1x2 = (2m-m ^ 2) / (2m-1)
Focus f (- 1,0) of a circle passing through an ellipse with diameter ab
So FA · FB = 0
（x1+1)(x2+1)+y1y2=0
(x1+1)(x2+1)+(x1-1)(x2-1)=0
x1x2=-1
So (2m-m ^ 2) / (2m-1) = - 1
m=2±√3
And because m > 1
So m = 2 + √ 3