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 (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 .

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 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 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.

If (1+ tan theta) (1+ tan phi)=2, then the value of (theta+ phi) is

If t_1a n dt_2 are the ends of a focal chord of the parabola y^2=4a x , then prove that the roots of the equation t_1x^2+a x+t_2=0 are real.

Let P(a sec theta , b tan theta ) and Q(a sec phi , b tan phi) where theta + phi = (pi)/(2) be two point on the hyperbola (x^(2))/(a^(2)) - (y^(2))/(b^(2)) =1 .If ( h, k) be the point of intersection of the normals at P and Q , then the value of k is -

If (acosthetasecphi-x)/(asin(theta+phi))=(y-bsinthetasecphi)/(bcos(theta+phi))=tanphi show that x^2/a^2+y^2/b^2=1

If alpha and beta be the eccentric angles of the extremities of a focal chord of the hyperbola b^(2)x^(2) - a^(2)y^(2) = a^(2)b^(2) , show that, tan(alpha)/(2)tan(beta)/(2) = -(e-1)/(e+1) , (e-1)/(e+1) (e is the eccentricity of the hyperbola.).

If (a^(2)-b^(2))sin theta+2 ab cos theta-(a^(2)+b^(2))=0 , then prove that tan theta=(1)/(2)((a)/(b)-(b)/(a)) .