Prove that the portion of the tangent to an ellipse intercepted between the ellipse and the directrix subtends a right angle at the corresponding focus.

Prove that the portion of the tangent intercepted,between the point of contact and the directrix of the parabola y^(2)= 4ax subtends a right angle at its focus.

Statement 1: In the parabola y^2=4a x , the circle drawn the taking the focal radii as diameter touches the y-axis. Statement 2: The portion of the tangent intercepted between the point of contact and directrix subtends an angle of 90^0 at focus.

Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre.

The exhaustive set of values of alpha^2 such that there exists a tangent to the ellipse x^2+alpha^2y^2=alpha^2 and the portion of the tangent intercepted by the hyperbola alpha^2x^2-y^2=1 subtends a right angle at the center of the curves is:

For the curve x y=c , prove that the portion of the tangent intercepted between the coordinate axes is bisected at the point of contact.

For the curve x y=c , prove that the portion of the tangent intercepted between the coordinate axes is bisected at the point of contact.

Prove that the normal chord to a parabola at the point whose ordinate is equal to the abscissa subtends a right angle at the focus.

Prove that the normal chord to a parabola at the point whose ordinate is equal to the abscissa subtends a right angle at the focus.

The exhaustive set value of alpha^2 such that there exists a tangent to the ellipse x^2+alpha^2y^2=alpha^2 and the portion of the tangent intercepted by the hyperbola alpha^2x^2-y^2 subtends a right angle at the center of the curves is (a) [(sqrt(5)+1)/2,2] (b) (1,2] (c) [(sqrt(5)-1)/2,1] (d) [(sqrt(5)-1)/2,1]uu[1,(sqrt(5)+1)/2]

Two parallel tangents of a circle meet a third tangent at points P and Q. Prove that PQ subtends a right angle at the centre.