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.

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)))

Two perpendicular tangents drawn to the ellipse (x^2)/(25)+(y^2)/(16)=1 intersect on the curve.

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

The sum of the squares of the perpendiculars on any tangents to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 from two points on the minor axis each at a distance a e from the center is 2a^2 (b) 2b^2 (c) a^2+b^2 a^2-b^2

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 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.

A point P moves such that the chord of contact of the pair of tangents from P on the parabola y^2=4a x touches the rectangular hyperbola x^2-y^2=c^2dot Show that the locus of P is the ellipse (x^2)/(c^2)+(y^2)/((2a)^2)=1.

Find the maximum area of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 which touches the line y=3x+2.

Let d be the perpendicular distance from the centre of the ellipse x^2/a^2+y^2/b^2=1 to the tangent drawn at a point P on the ellipse. If F_1 & F_2 are the two foci of the ellipse, then show the (PF_1-PF_2)^2=4a^2[1-b^2/d^2] .