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.

The tangent PT and the normal PN to the parabola y^2=4ax at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose:

Tangents are drawn to the parabola y^2=4a x at the point where the line l x+m y+n=0 meets this parabola. Find the point of intersection of these tangents.

Tangents are drawn to the hyperbola 4x^2-y^2=36 at the points P and Q. If these tangents intersect at the point T(0,3) then the area (in sq units) of triangle PTQ is

Tangent are drawn from the point (-1,2) on the parabola y^2=4x . Find the length that these tangents will intercept on the line x=2.

From a point on the circle x^2+y^2=a^2 , two tangents are drawn to the circle x^2+y^2=b^2(a > b) . If the chord of contact touches a variable circle passing through origin, show that the locus of the center of the variable circle is always a parabola.

A tangent is drawn to the parabola y^2=4 x at the point P whose abscissa lies in the interval (1, 4). The maximum possible area of the triangle formed by the tangent at P , the ordinates of the point P , and the x-axis is equal to

The locus of the point of intersection of perependicular tangent of the parabola y^(2) =4ax is

Find the locus of the point from which the two tangents drawn to the parabola y^2=4a x are such that the slope of one is thrice that of the other.

The tangent at any point P onthe parabola y^2=4a x intersects the y-axis at Qdot Then tangent to the circumcircle of triangle P Q S(S is the focus) at Q is

Two tangent are drawn from the point (-2,-1) to parabola y^2=4xdot if alpha is the angle between these tangents, then find the value of tanalphadot