Prove that the chord of contact of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` with respect to any point on the directrix is a focal chord.

The locus of the point which is such that the chord of contact of tangents drawn from it to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 forms a triangle of constant area with the coordinate axes is (a) straight line (b) a hyperbola (c) an ellipse (d) a circle

Prove that the chords of contact of pairs of perpendicular tangents to the ellipse x^2/a^2+y^2/b^2=1 touch another fixed ellipse.

show that the length of the focal chord of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 which makes an angle theta with the major axis is (2ab^(2))/(a^(2) sin ^(2) theta+ b^(2) cos^(2) theta) unit.

Show that the tangents at the end of any focal chord of the ellipse x^(2)b^(2)+y^(2)a^(2)=a^(2)b^(2) intersect on the directrix.

A O B is the positive quadrant of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 which has O A=a ,O B=b . Then find the area between the arc A B and the chord A B of the ellipse.

Show that the midpoints of focal chords of a hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 lie on another similar hyperbola.

Find the locus of middle points of chords of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 which subtend right angle at its center.

If the tangent at any point of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 makes an angle alpha with the major axis and an angle beta with the focal radius of the point of contact, then show that the eccentricity of the ellipse is given by e=cosbeta/(cosalpha)

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.

Find the locus of middle points of chords of the ellipse x^2/a^2+y^2/b^2=1 which subtend right angles at its centre.