If the chord joining the points `(asectheta, btantheta)`and `(asecphi, btanphi)` on the hyperbola `x^2/a^2-y^2/b^2=1` passes through the focus (ae,0), prove that `tan(theta/2)tan(phi/2)+(e-1)/(e+1)=0`.

If (asectheta;btantheta) and (asecphi; btanphi) are the ends of the focal chord of x^2/a^2-y^2/b^2=1 then prove that tan(theta/2)tan(phi/2)=(1-e)/(1+e)

If the chord joining the points P(theta)' and 'Q(phi)' of the ellipse x^2/a^2+y^2/b^2=1 subtends a right angle at (a,0) prove that tan(theta/2)tan(phi/2)=-b^2/a^2 .

If the chord joining points P(alpha) and Q(beta) on the ellipse ((x^2)/(a^2))+((y^2)/(b^2))=1 subtends a right angle at the vertex A(a ,0), then prove that tan(alpha/2)tan(beta/2)=-(b^2)/(a^2)dot

The slop of the normal to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at the point ( a sec theta , b tan theta) is -

Let P(a sectheta, btantheta) and Q(asecphi , btanphi) (where theta+phi=pi/2 ) be two points on the hyperbola x^2/a^2-y^2/b^2=1 If (h, k) is the point of intersection of the normals at P and Q then k is equal to (A) (a^2+b^2)/a (B) -((a^2+b^2)/a) (C) (a^2+b^2)/b (D) -((a^2+b^2)/b)

If e is the eccentricity of the hyperbola (x^(2))/(a^(2)) - (y^(2))/(b^(2)) = 1 , then e =

If sec theta+tan theta = x , prove that sin theta= (x^2-1)/(x^2+1)

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

The tangent to the curve y=xe^(x^2) passing through the point (1,e) also passes through the point

If theta and phi are the eccentric angles of the end points of a chord which passes through the focus .of an ellipse x^2/a^2+y^2/b^2=1 .Show that tan(0//2)tan(phi//2)=((e-1)/(e+1)) ,where e is the eccentricity of the ellipse.