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 .

Find the eccentricity of the ellipse if the length of rectum is equal to half the minor axis of the ellipse .

Find the locus of the middle points of chords of an ellipse drawn through the positive extremity of the minor axis

If (5, 12) and (24, 7) are the foci of an ellipse passing through the origin, then find 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

From any point on any directrix of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1,a > b , a pari of tangents is drawn to the auxiliary circle. Show that the chord of contact will pass through the correspoinding focus of the ellipse.

P is the point on the ellipse is x^2/16+y^2/9=1 and Q is the corresponding point on the auxiliary circle of the ellipse. If the line joining the center C to Q meets the normal at P with respect to the given ellipse at K, then find the value of CK.

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 .

If the length of the minor axis of an ellipse is equal to the distance between their foci, then eccntricity of the ellipse is _

The co-ordinate of a point on the auxiliary circle of the ellipse x^(2)+2y^(2)=4 corresponding to the point on the ellipse whose eccentric angle is 60^(@) will be

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 .