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.

In a A B C , if cosC=(sinA)/(2sinB) , prove that the triangle is isosceles.

If atanA+btanB=(a+b)tan((A+B)/2) prove that triangle ABC is isosceles.

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.

Find the eccentricity of the ellipse if y=x and 3y+2x=0 are the equations of a pair of its conjugate diameters.

Consider the hyperbola x^(2)/a^(2) - y^(2)/b^(2) = 1 . Area of the triangle formed by the asymptotes and the tangent drawn to it at (a, 0) is

In a rectangular hyperbola x^(2)-y^(2)=a^(2) , prove that SP*S'P=CP^(2) where P is any point on the hyperbola, C is origin and S and S' are foci.

In DeltaABC, (a^(2)+b^(2))/(a^(2)-b^(2))=(sin(A+B))/(sin(A-B)) , prove that the triangle is isosceles or right triangle.

An isosceles right triangle has area 8 cm^(2) . The length of its hypotenuse is

The hyperbola x^(2)/a^(2)-y^(2)/b^(2)=1 has its conjugate axis 5 and passes through the point (2,-1). Then find the length of the latus rectum.

The asymptote of the hyperbola x^2/a^2-y^2/b^2=1 form with an tangent to the hyperbola triangle whose area is a^2 tan lambda in magnitude then its eccentricity is: (a) sec lambda (b) cosec lambda (c) sec^2 lambda (d) cosec^2 lambda