[" If "P(a sec alpha,b tan alpha)" and "Q(a sec beta)" are two points on the hyperbola "(x^(2))/(a^(2))-(y^(2))/(b^(2))=1." Such that "alpha-beta=2 theta" ,"],[" then "PQ" touches the hyperbola "]

If P(a sec alpha,b tan alpha) and Q(a secbeta, b tan beta) are two points on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 such that alpha-beta=2theta (a constant), then PQ touches the hyperbola

If P(a sec alpha,b tan alpha) and Q(a secbeta, b tan beta) are two points on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 such that alpha-beta=2theta (a constant), then PQ touches the hyperbola

P( asec alpha, b tan alpha ) and Q (a sec beta, b tan beta) are two given points on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2)) = 1. Show that the equation of the chord PQ is (x)/(a) cos (alpha - beta)/(2) - (y)/(b) sin (alpha + beta)/(2) = cos (alpha+beta)/(2).

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

If alpha and beta are two points on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 and the chord joining these two points passes through the focus (ae,0) then e cos((alpha-beta)/(2))

If (a sec alpha,b tan alpha) and (a sec beta ,b tan beta) be the ends of a chord of (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 passing through the focus (ae,0) then tan (alpha/2) tan (beta/2) is equal to

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 =