Prove that perpendicular drawn from focus upon any tangent of a parabola lies on the tangent at the vertex

Prove that in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

Prove that the locus of the foot of the perpendicular drawn from the focus of the parabola y ^(2) = 4 ax upon any tangent to its is the tangent at the vertex.

If S be the focus and SH be perpendicular to the tangent at P , then prove that H lies on the tangent at the vertex and SH^2 = OS.SP , where O is the vertex of the parabola.

The locus of foot of the perpendiculars drawn from the focus on a variable tangent to the parabola y^2 = 4ax is

Prove that the product of the perpendicular from the foci on any tangent to an ellipse is equal to the square of the semi-minor axis.

The product of the perpendiculars drawn from the two foci of an ellipse to the tangent at any point of the ellipse is

The locus of the foot of perpendicular from my focus of a hyperbola upon any tangent to the hyperbola is the auxiliary circle of the hyperbola. Consider the foci of a hyperbola as (-3, -2) and (5,6) and the foot of perpendicular from the focus (5, 6) upon a tangent to the hyperbola as (2, 5). The point of contact of the tangent with the hyperbola is

The locus of the foot of perpendicular from my focus of a hyperbola upon any tangent to the hyperbola is the auxiliary circle of the hyperbola. Consider the foci of a hyperbola as (-3, -2) and (5,6) and the foot of perpendicular from the focus (5, 6) upon a tangent to the hyperbola as (2, 5). The directrix of the hyperbola corresponding to the focus (5, 6) is

The locus of the foot of perpendicular from my focus of a hyperbola upon any tangent to the hyperbola is the auxiliary circle of the hyperbola. Consider the foci of a hyperbola as (-3, -2) and (5,6) and the foot of perpendicular from the focus (5, 6) upon a tangent to the hyperbola as (2, 5). The conjugate axis of the hyperbola is

If SY and S'Y' be drawn perpendiculars from foci to any tangent to a hyperbola. Prove that y and Y' lie on the auxiliary circle and that product of these perpendicular is constant.