Hyperbola

Chords of the hyperbola $$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$$ are tangents to the circle drawn on the line joining the foci as diameter. Find the locus of the point of intersection of the tangent at the extremities of the chords.
If $$C$$ is centre of a hyperbola $$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$$, $$S, S'$$ its foci and $$P$$ a point on it. Then $$SP. S'P = CP^2 - a^2 + b^2$$.
Tangents are drawn to a hyperbola from a point on one of the branches of its conjugate hyperbola. Show that chord of contact will touch the other branch of the conjugate hyperbola.
A straight line is drawn parallel to the conjugate axis of the hyperbola $$\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$$ to meet it and the conjugate hyperbola respectively in the point $$P$$ and $$Q$$. The normals at $$p$$ and $$Q$$ to the curves meet on 
A point $$P$$ moves such that sum of the slopes of the normals drawn from it to the hyperbola $$xy=16$$ is equal to the sum of ordinates of feet of normals. The locus of $$P$$ is a curve $$C$$.