The centroid of the triangle formed by the feet of three normals to the parabola `y^2=4ax`

The algebraic sum of the ordinates of the feet of 3 normals drawn to the parabola y^2=4ax from a given point is 0.

(i) Tangents are drawn from the point (alpha, beta) to the parabola y^2 = 4ax . Show that the length of their chord of contact is : 1/|a| sqrt((beta^2 - 4aalpha) (beta^2 + 4a^2)) . Also show that the area of the triangle formed by the tangents from (alpha, beta) to parabola y^2 = 4ax and the chord of contact is (beta^2 - 4aalpha)^(3/2)/(2a) . (ii) Prove that the area of the triangle formed by the tangents at points t_1 and t_2 on the parabola y^2 = 4ax with the chord joining these two points is a^2/2 |t_1 - t_2|^3 .

Area of the triangle formed by the tangents from (x_1,y_1) to the parabola y^2 = 4 ax and its chord of contact is (y_1^2-4ax_1)^(3/2)/(2a)=S_11^(3/2)/(2a)

STATEMENT-1 :The locus of centroid of a triangle formed by three co-normal points on a parabola is the axis of parabola. STATEMENT-2 : One of the angles between the parabolas y^(2) =8x and x^(2) = 27y is tan^(-1)((9)/(13)). STATEMENT-3 : Consider the ellipse (x^(2))/(9) + (y^(2))/(4) =1 THe product of lengths of perpendiculars drawn from foci to a tangent is 4.

The area of triangle formed by tangent and normal at (8, 8) on the parabola y^(2)=8x and the axis of the parabola is

The locus of centroid of triangle formed by a tangent to the parabola y^(2) = 36x with coordinate axes is (a) y^(2) =- 9x (b) y^(2) +3x = 0 (c) y^(2) = 3x (d) y^(2) = 9x

The centroid of the triangle formed by (2, -5), (2, 7), (4, 7) is

Length of the shortest normal chord of the parabola y^2=4ax is

The centroid of the triangle formed by (7, 4), (4, -6), (-5, 2) is

The locus of the point of intersection of tangents drawn at the extremities of a normal chord to the parabola y^2=4ax is the curve