It is known that the eccentricity of the ellipse x2 / A2 + Y2 / B2 = 1 (a > b > 0) is √ 2 / 2, and the right focus is f (1,0) If a straight line with an inclination angle of 45 ° intersects the ellipse at two points a and B, calculate the value of ab

It is known that the eccentricity of the ellipse x2 / A2 + Y2 / B2 = 1 (a > b > 0) is √ 2 / 2, and the right focus is f (1,0) If a straight line with an inclination angle of 45 ° intersects the ellipse at two points a and B, calculate the value of ab

The focal half distance C = 1
Then E = C / a = √ 2 / 2, then a = √ 2 · C = √ 2, then B = √ (A & # 178; - C & # 178;) = 1
Then the elliptic equation is: X & # 178; + 2Y & # 178; = 2
The slope of the line with 45 ° inclination angle is k = 1, and the line passing through f (1,0) is y = K (x-1)
Then the linear equation is y = X-1
Substituting y = X-1 into X & # 178; + 2Y & # 178; = 2, we get X & # 178; + 2 (x-1) &# 178; = 2, that is, 3x & # 178; - 4x = 0, that is, X (3x-4) = 0
Then X1 = 0, X2 = 4 / 3
Substituting y = X-1, the two intersections are (0, - 1), (4 / 3,1 / 3) respectively
Then | ab | = √ [(0-4 / 3) & # 178; + (- 1-1 / 3) & # 178;] = 4 √ 2 / 3