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.

Sum of the focal distance of the ellipse (x^2)/(a^2) + (y^2)/(b^2) = 1 is

Prove that the chords of constant of perpendicular tangents to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 touch another fixed ellipse (x^(2))/(a^(4))+(y^(2))/(b^(4))=(1)/((a^(2)+b^(2)))

The locus of mid-points of a focal chord of the ellipse x^2/a^2+y^2/b^2=1

Statement 1 : Tangents are drawn to the ellipse (x^2)/4+(y^2)/2=1 at the points where it is intersected by the line 2x+3y=1 . The point of intersection of these tangents is (8, 6). Statement 2 : The equation of the chord of contact to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 from an external point is given by (x x_1)/(a^2)+(y y_1)/(b^2)-1=0

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 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.

If the midpoint of the chord of the ellipse x^2/16+y^2/25=1 is (0,3)

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

Statement 1 : If from any point P(x_1, y_1) on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 , tangents are drawn to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1, then the corresponding chord of contact lies on an other branch of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 Statement 2 : From any point outside the hyperbola, two tangents can be drawn to the hyperbola.

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)