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

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 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 =

For the hyperbola (x^(2))/(cos^(2) alpha ) - (y^(2))/(sin^(2) alpha ) = 1 , where alpha is a parameter, which of the following remains constant ?

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) .

Prove that the locus of the middle-points of the chords of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 which pass through a fixed point (alpha, beta) is a hyperbola whose centre is ((alpha)/(2), (beta)/(2)) .

A chord PQ of the hyperbola xy=c^(2) is tangent to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 Find the locus of the middle point of PQ .

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))

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