If any tangent to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` with centre C, meets its director circle in P and Q, show that CP and CQ are conjugate semi-diameters of the hyperbola.

Let P(6, 3) be a point on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1. If the normal at point P intersects the x-axis at (9, 0), then find the eccentricity of the hyperbola.

For the hyperbola x^(2)-y^(2)=a^(2) , prove that the triangle CPD is isosceles and has constant area, where CP and CD are a pair of its conjugate diameter.

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.

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.Locus of point C is

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

For the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 , let n be the number of points on the plane through which perpendicular tangents are drawn.

If it is possible to draw the tangent to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 having slope 2, then find its range of eccentricity.

The locus of the poles of the chords of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 which subtend a right angle at its centre is

The ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 and the hyperbola (x^(2))/(A^(2))-(y^(2))/(B^(2))=1 are given to be confocal and length of mirror axis of the ellipse is same as the conjugate axis of the hyperbola. If e_1 and e_2 represents the eccentricities of ellipse and hyperbola respectively, then the value of e_(1)^(-2)+e_(1)^(-2) is