Locus of foot of perpendicular from focus upon any tangent is tangent at vertex

The locus of feet of perpendiculars from the focii upon any tangent is an auxilliary circle.

Let the two foci of an ellipse be (-1,0) and (3,4) and the foot of perpendicular from the focus (3,4) upon a tangent to the ellipse be (4,6). The foot of perpendicular from the focus (-1,0) upon the same tangent to the ellipse is

The locus of the foot of the perpendicular from the foci an any tangent to the ellipse x^(2)/a^(2) + y^(2)/b^(2) = 1 , is

If the locus of the foot of the perpendicular drawn from centre upon any tangent to the ellipse (x^(2))/(40)+(y^(2))/(10)=1 is (x^(2)+y^(2))^(2)=ax^(2)+by^(2) , then (a-b) is equal to

Show that the locus of the foot of the perpendicular drawn from focus to a tangent to the hyperbola x^2/a^2 - y^2/b^2 = 1 is x^2 + y^2 = a^2 .

Let S=(3,4) and S'=(9,12) be two foci of an ellipse.If the coordinates of the foot of the perpendicular from focus S to a tangent of the ellipse is (1,-4) then the eccentricity of the ellipse is