PQ is a chord joining the points `phi_1` and `phi_2` on the hyperbola `x^2/a^2 - y^2/b^2 = 1`. If `phi_1 and phi_2 = 2 alpha`, where`alha` is constant, prove that PQ touches the hyperbola `x^2/a^2 cos^2 alpha - y^2 /b^2 = 1`

PQ is a chord joining the points phi_(1) and phi_(2) on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1. If phi_(1) and phi_(2)=2 alpha, wherealha is constant, prove that PQ touches the hyperbola (x^(2))/(a^(2))cos^(2)alpha-(y^(2))/(b^(2))=1

The line x cos alpha + y sin alpha = p touches the hyperbola (x ^(2))/( a ^(2)) - (y ^(2))/( b ^(2))= 1 , if :

If the chords through point theta and phi on the ellipse x^2/a^2 + y^2/b^2 = 1 . Intersect the major axes at (c,0).Then prove that tan theta/2 tan phi/2 = (c-a)/(c+a) .

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 through the points (a sec theta, b tan theta) and (a sec phi, b tan phi) on the hyperbola x^2/a^2 - y^2/b^2 = 1 passes through a focus, prove that tan (theta/2) tan (phi/2) + (e-1)/(e+1) = 0 .

If the chord through the points (a sec theta, b tan theta) and (a sec phi, b tan phi) on the hyperbola x^2/a^@ - y^2/b^2 = 1 passes through a focus, prove that tan theta/2 tan phi/2 + (e-1)/(e+1) = 0 .

If a chord PQ , joining P (theta) " Q and " (phi) , of an ellipse (x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 , subtands a right angle at its centre , then tan theta * tan phi =

Let A (2sec theta, 3 tan theta ) and B( 2sec phi ,3 tan phi ) where theta + phi =(pi)/(2) be two point on the hyperbola (x^(2))/( 4) -( y^(2))/( 9) =1 . If (alpha , beta ) is the point of intersection of normals to the hyperbola at A and B ,then beta =

If the line y=mx + sqrt(a^2 m^2 - b^2) touches the hyperbola x^2/a^2 - y^2/b^2 = 1 at the point (a sec phi, b tan phi) , show that phi = sin^(-1) (b/(am)) .