It is known that F1F2 is the focus of the ellipse x2 / A2 + Y2 / B2 = 1 (a > b > 0), the point P is on the ellipse, and the angle f1pf2 = 90 °, the intersection of the line segment Pf1 and the axis is Q, and O is the origin of the coordinate. If the area ratio of the triangle f1oq to the quadrilateral of2pq is 1:2, the eccentricity of the ellipse is 0

It is known that F1F2 is the focus of the ellipse x2 / A2 + Y2 / B2 = 1 (a > b > 0), the point P is on the ellipse, and the angle f1pf2 = 90 °, the intersection of the line segment Pf1 and the axis is Q, and O is the origin of the coordinate. If the area ratio of the triangle f1oq to the quadrilateral of2pq is 1:2, the eccentricity of the ellipse is 0

Let p be in the first quadrant (the same as other quadrants), then PQ: QF1 = 1:2,
P(c/2,√3c/2),PF1=√3c,PF2=c,PF1+PF2=2a,e=√3-1.

Given that P (x, y) is any point on the curve X & # 178; + (Y-2) &# 178; = 3, two methods are needed to find the maximum value of 2x + y

x²+(y-2)²=3
∴ x=√3cosA,y=2+√3sinA
∴ 2x+y=2√3cosA+2+√3sinA=√15sin(A+∅)+2
The maximum value is √ 15 + 2
Let 2x + y = t, a line and a circle have a common point
If it is a straight line, then the distance from the center of the circle (0,2) to the straight line ≤ radius
∴ d=|2-t|/√5≤√3
∴ |t-2|≤√15
∴ -√15+2≤t≤2+√15
The maximum value of 2x + y is √ 15 + 2

Given that P (x, y) is the point of the curve X = 1 + cos θ, y = √ 3sin θ, then the maximum value of X & # 178; + Y & # 178; is

X & # 178; + Y & # 178; = (1 + cos θ) & # 178; + 3sin & # 178; θ = 1 + 2cos θ + cos & # 178; θ + 3 (1-cos & # 178; θ) = - 2cos & # 178; θ + 2cos θ + 4 = - 2 (COS θ - 1 / 2) & # 178; + 9 / 2 when cos θ = 1 / 2, X & # 178; + Y & # 178; achieves the maximum value of 9 / 2