Show that the tangents at the end of any focal chord of the ellipse `x^(2)b^(2)+y^(2)a^(2)=a^(2)b^(2)` intersect on the directrix.

Show that the minimum value of the length of a tangent to the ellipse b^(2)x^(2)+a^(2)y^(2)=a^(2)b^(2) intercepted between the axes is (a+b).

show that the length of the focal chord of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 which makes an angle theta with the major axis is (2ab^(2))/(a^(2) sin ^(2) theta+ b^(2) cos^(2) theta) unit.

Prove that the locus of the point of intersection of the tangents at the ends of the normal chords of the hyperbola x^(2)-y^(2)=a^(2)" is " a^(2)(y^(2)-x^(2))=4x^(2)y^(2) .

Show tht the locus of the middle points of chords of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) = 1 drawn through an extremity of the minor axis is (x^(2))/(a^(2))+(y^(2))/(b^(2)) = pm (y)/(b)

Show that the product of the ordinates of the ends of a focal chord of the parabola y^(2) = 4ax is constant .

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

Find the slope of normal to the ellipse , (x^(2))/(a^(2))+(y^(2))/(b^(2))=2 at the point (a,b).

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.

Prove that the locus of the point of intersection of the tangents at the ends of the normal chords of the hyperbola x^2-y^2=a^2 is a^2(y^2-x^2)=4x^2y^2dot

If the tangent at any point on the ellipse (x^(2))/(a^(2)) + (y^(2))/(b^(2)) =1 intersects the coordinate axes at P and Q , then the minimum value of the area (in square unit ) of the triangle OPQ is (O being the origin )-