Question

已知中心在原点O,焦点在x轴上的椭圆E过点(0,1),离心率为根号2/2.1.求椭圆E的方程,这个会,第二问?

Answer 1

= 1, c/a = sqrt(2)/2

c = sqrt(a^2 - b^2) = sqrt(a^2 - 1)

c^2/a^2 = 1-1/a^2 = 1/2

so a = sqrt(2), c=1

x^2/2 + y^2 = 1

2. equation of line L is y = k(x+1)

its intersecting points with E are obtained by solving

x^2 + 2k^2 (x+1)^2 - 2 = 0

x = (-2k^2 +/- sqrt(2k^2 - 2)/(2k^2 + 1)

you can get A,B's coordinate this way, and therefore C, and then M

area of MAF is a function of k

once k is obtained, M,A,F,C's coordinates are known

and you can get the equation of the circle