Find the area of the triangle formed by any tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` with its asymptotes.

Prove that the area of the triangle formed by any tangent to the hyperbola xy=c^(2) and the coordinate axes is constant.

Find the equations of the tangent and normal to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at the point (x0, y0).

Find the equation of the tangent to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at ( a sec, theta b tan theta) . Hence show that if the tangent intercepts unit length on each of the coordinate axis than the point (a,b) satisfies the equation x^(2)-y^(2)=1

The slope of the tangent to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at the point ( x_(1),y_(1)) is-

Find the product of the length of perpendiculars drawn from any point on the hyperbola x^2-2y^2-2=0 to its asymptotes.

Find the angle between the asymptotes of the hyperbola (x^2)/(16)-(y^2)/9=1 .

Find the area of the triangle formed by the lines joining the vertex of the parabola x^(2)=12y to the ends of its latus rectum.

Find the area of the triangle formed by the lines joingn the vertex of the parabola x^2=24y to the ends of its latus rectum .

If the latus rectum subtends a right angle at the center of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 , then find its eccentricity.

Find the equations to the common tangents to the two hyperbolas (x^2)/(a^2)-(y^2)/(b^2)=1 and (y^2)/(a^2)-(x^2)/(b^2)=1