If the normal at one end of lotus rectum of an ellipse passes through one end of minor axis then prove that,
`e^(2)=(sqrt(5)-1)/(2)`, where e is the eccentricity of the ellipse.

If the normal at one end of lotus rectum of an ellipse passes through one end of minor axis then prove that, e^(4)+e^(2)-1=0

If the normal at one end of the latus rectum of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 passes through one end of the minor axis, then prove that eccentricity is constant.

Show that the tangents at the ends of latus rectum of an ellipse intersect on the major axis.

If theta and phi are the eccentric angles of the end points of a chord which passes through the focus .of an ellipse x^2/a^2+y^2/b^2=1 .Show that tan(0//2)tan(phi//2)=((e-1)/(e+1)) ,where e is the eccentricity of the ellipse.

The latus rectum of an ellipse is equal to one-half of its minor axis. The eccentricity of the ellipse is

S and T are the foci of an ellipse and B is the end point of the minor axis. If STB is equilateral triangle, the eccentricity of the ellipse is

If the angle between the lines joining the end points of minor axis of an elipes with its one focus is pi/2, then the eccentricity of the ellipse is-

Chords of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 are drawn through the positive end of the minor axis. Then prove that their midpoints lie on the ellipse.

An ellipse and a hyperbola are confocal (have the same focus) and the conjugate axis of the hyperbola is equal to the minor axis of the ellipse. If e_1a n de_2 are the eccentricities of the ellipse and the hyperbola, respectively, then prove that 1/(e_1^2)+1/(e_2^2)=2 .

The auxiliary circle of a family of ellipses passes through the origin and makes intercepts of 8 units and 6 units on the x and y-axis, respectively. If the eccentricity of all such ellipses is 1/2, then find the locus of the focus.