Tangents to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` make angle `theta_(1), theta_(2)` with transvrse axis of a hyperbola. Show that the points of intersection of these tangents lies on the curve `2xy=k(x^(2)-a^(2))` when `tan theta_(1)+ tan theta_(2)=k`

Find the locus of the points of the intersection of tangents to ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 which make an angle theta.

If the tangents drawn from a point on the hyperbola x^(2)-y^(2)=a^(2)-b^(2) to ellipse (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 make angle alpha and beta with the transverse axis of the hyperbola, then

Find the equation of the tangent to the hyperbola 4x^(2)-9y^(2)=36" at "theta=pi/4

Two tangents to the parabola y^2 = 4ax make angles theta_1,theta_2 with the x-axis. Then the locus of their point of intersection if cot theta_1 + cot theta_2=c is

If e is the eccentricity of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 and theta is the angle between the asymptotes, then cos.(theta)/(2) is equal to

If theta _(1), theta_(2) are the angles of inclination of tangents through a point P to the circle x^(2)+y^(2) = a^(2) then find the locus of P when cot theta_(1) + cot theta_(2) = k.

The point of intersection of two tangents to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 , the product of whose slopes is c^(2) , lies on the curve

The locus of the point of intersection of two tangents to the hyperbola x^(2)//a^(2) -y^(2)//b^(2) = 1 which make an angle 90^(@) with one another is

The acute angle of intersection of the curves x^(2)y=1 and y=x^(2) in the first quadrant is theta , then tan theta is equal to

Let P(theta_1) and Q(theta_2) are the extremities of any focal chord of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 whose eccentricity is e. Let theta be the angle between its asymptotes. Tangents are drawn to the hyperbola at some arbitrary points R. These tangent meet the coordinate axes at the points A and B respectively. The rectangle OABC (O being the origin) is completedm, then Q. If cos^(2)((theta_1+theta_2)/(2))=lambdacos^(2)((theta_1-theta_2)/(2)) , then lambda is equal to