Tangents are drawn to the parabola at three distinct points. Prove that these tangent lines always make a triangle and that the locus of the orthocentre of the triangle is the directrix of the parabola.

Tangents are drawn to the parabola y^2=12x at the points A, B and C such that the three tangents form a triangle PQR. If theta_1,theta_2 and theta_3 be the inclinations of these tangents with the axis of x such that their cotangents form an arithmetical progression (in the same order) with common difference 2. Find the area of the triangle PQR.

The locus of the Orthocentre of the triangle formed by three tangents of the parabola (4x-3)^(2)=-64(2y+1) is

If the tangents at two points (1,2) and (3,6) on a parabola intersect at the point (-1,1), then ths slope of the directrix of the parabola is

Prove that BS is perpendicular to PS,where P lies on the parabola y^(2)=4ax and B is the point of intersection of tangent at P and directrix of the parabola.

The normals at point A and B on y^(2)=4ax meet the parabola at C on the same parabola then the locus of orthocenter of triangle ABC is