A tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` cuts the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` at `Pa n dQ` . Show that the locus of the midpoint of `P Q` is `((x^2)/(a^2)+(y^2)/(b^2))^2=(x^2)/(a^2)-(y^2)/(b^2)dot`

Length of common tangents to the hyperbolas (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 and (y^(2))/(a^(2))-(x^(2))/(b^(2))=1 is

For the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 and (x^(2))/(b^(2))+(y^(2))/(a^(2)) =1

The tangent at P on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 meets the asymptotes -(y)/(a)-(y^(2))/(b)=0 if the locus of the midpoint of PQ has the equation (x^(2))/(a^(2))-(y^(2))/(b^(2))=k, then k has the value equal to

Tangents at right angle are drawn to the ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 . Show that the focus of the middle points of the chord of contact is the curve (x^(2)/a^(2)+y^(2)/b^(2))^(2)=(x^(2)+y^(2))/(a^(2)+b^(2)) .

The area of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 is

A tangent is drawn to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 to cut the ellipse (x^(2))/(c^(2))+(y^(2))/(d^(2))=1 at the points P and Q . If tangents at P and Q to the ellipse (x^(2))/(c^(2))+(y^(2))/(d^(2))=1 intersect at right angle then prove that (a^(2))/(c^(2))+(b^(2))/(d^(2))=1

A tangent (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 meets the axes at A and B.Then the locus of mid point of AB is