From any point of the hyperbola `x^(2)/a^(2)-y^(2)/b^(2)=1`, tangents are drawn to another hyperbola which has the same asymptotes. Show that the chord of con- tact cuts off a constant area from the asymptotes.

From any point to the hyperbola :.(2)/(a^(2))-(y^(2))/(b^(2))=1, tangents are drawn to thehyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=2 The area cut off bythe chord of contact on the regionbetween the asymptotes is equal to

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.

From any point on the hyperbola H_(1):(x^2//a^2)-(y^2//b^2)=1 tangents are drawn to the hyperbola H_(2): (x^2//a^2)-(y^2//b^2)=2 .The area cut-off by the chord of contact on the asymp- totes of H_(2) is equal to

From points on the circle x^2+y^2=a^2 tangents are drawn to the hyperbola x^2-y^2=a^2 . Then, the locus of mid-points of the chord of contact of tangents is:

Show that there cannot be any common tangent to the hyperbola x^(2)/a^(2) - y^(2)/b^(2) = 1 and its conjugate hyperbola.

Consider the hyperbola x^(2)/a^(2) - y^(2)/b^(2) = 1 . Area of the triangle formed by the asymptotes and the tangent drawn to it at (a, 0) is

tangent is drawn at point P(x_(1),y_(1)) on the hyperbola (x^(2))/(4)-y^(2)=1. If pair of tangents are drawn from any point on this tangent to the circle x^(2)+y^(2)=16 such that chords of contact are concurrent at the point (x_(2),y_(2)) then