The locus of the mid points of the chords of the hyperbola x² $$-$$ y² = 4, which touch the parabola y² = 8x, is :

The locus of the centroid of the triangle formed by any point P on the hyperbola $$16{x^2} - 9{y^2} + 32x + 36y - 164 = 0$$, and its foci is :

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 :