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.

The tangent at a point P(acosvarphi,bsinvarphi) of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 meets its auxiliary circle at two points, the chord joining which subtends a right angle at the center. Find the eccentricity of the ellipse.

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.

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.

PA and PB are tangents drawn from a point P to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 . The area of the triangle formed by the chord of contact AB and axes of co-ordinates is constant. Then locus of P is:

The tangent at a point P on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 , which in not an extremely of major axis meets a directrix at T. Statement-1: The circle on PT as diameter passes through the focus of the ellipse corresponding to the directrix on which T lies. Statement-2: Pt substends is a right angle at the focus of the ellipse corresponding to the directrix on which T lies.

The distance of the point 'theta' on the ellipse x^(2)/a^(2) + y^(2)/b^(2) = 1 from a focus, is

From a point on the axis of x common tangents are drawn to the parabola the ellipse y^(2) =4x and the ellipse x^(2)/a^(2)+y^(2)/b^(2)=1(agtbgt0) . If these tangents from an equilateral trianlge with their chord of contact w.r.t parabola, then set of exhaustive values of a is

From a point on the circle x^2+y^2=a^2 , two tangents are drawn to the circle x^2+y^2=b^2(a > b) . If the chord of contact touches a variable circle passing through origin, show that the locus of the center of the variable circle is always a parabola.

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.

Triangles are formed by pairs of tangent dreawn from any point on the ellipse a^(2)x^(2)+b^(2)y6(2)=(a^(2)+b^(2)^(2) to the ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 and the chord of contact. Show that the orthocentre of each such triangles lies triangle lies on the ellipse.