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 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 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.

Prove that the straight line joining the upper end of one latus rectum and lower end of other latus rectum of an ellipse passes through the centre of the ellipse .

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

Show that the double ordinate of the auxiliary circle of an ellipse passing through the focus is equal to the minor axis of the ellipse .

The length of minor axis of the ellipse 9x^(2) + 4y ^(2) = 36 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 .

Consider an ellipse having its foci at A(z_1)a n dB(z_2) in the Argand plane. If the eccentricity of the ellipse be e and it is known that origin is an interior point of the ellipse, then prove that e in (0,(|z_1-z_2|)/(|z_1|+|z_2|))