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)`.

If the line y = mx + sqrt(a^(2) m^(2) -b^(2)), m = (1)/(2) touches the hyperbola (x^(2))/(16)-(y^(2))/(3) =1 at the point (4 sec theta, sqrt(3) tan theta) then theta is

If the line lx+my+n=0 touches the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 . Then

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 .

Length of common tangents to the hyperbolas x^2/a^2-y^2/b^2=1 and y^2/a^2-x^2/b^2=1 is

Statement -1 : The lines y = pm b/a xx are known as the asymptotes of the hyperbola x^(2)/a^(2) - y^(2)/b^(2) - 1 = 0 . because Statement -2 : Asymptotes touch the curye at any real point .

The line 2x + y = 1 is tangent to the hyperbola x^2/a^2-y^2/b^2=1 . If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

The line 2x + y = 1 is tangent to the hyperbola x^2/a^2-y^2/b^2=1 . If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

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

If phi (x)=phi '(x) and phi(1)=2 , then phi(3) equals

Tangents are drawn from the point (alpha, 2) to the hyperbola 3x^2 - 2y^2 = 6 and are inclined at angles theta and phi to the x-axis . If tan theta. tan phi = 2, then the value of 2alpha^2 - 7 is