Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x² $$-$$ y² = 3. If L is also a tangent to the parabola y² = $$\alpha$$x, then $$\alpha$$ is equal to :

Consider a hyperbola H : x² $$-$$ 2y² = 4. Let the tangent at a
point P(4, $${\sqrt 6 }$$) meet the x-axis at Q and latus rectum at R(x₁, y₁), x₁ > 0. If F is a focus of H which is nearer to the point P, then the area of $$\Delta$$QFR is equal to :

The locus of the midpoints of the chord of the circle, x² + y² = 25 which is tangent to the hyperbola, $${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1$$ is :

A hyperbola passes through the foci of the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is :