The tangents to the parabola `y^2 4ax` at `P(at_1^2,2at_1)` and `Q(at_2^2,2at_2)` at R. Prove that the area of the `DeltaPQR` is `1/2a^2|(t_1-t_2)|^3`.

The tangents to the parabola y^2=4ax at P(at_1^2,2at_1) , and Q(at_2^2,2at_2) , intersect at R. Prove that the area of the triangle PQR is 1/2a^2(t_1-t_2)^3

The tangent to the parabola y^(2)=4ax at P(at_(1)^(2), 2at_(1))" and Q"(at_(2)^(2), 2at_(2)) intersect on its axis, them

Find the distance between the points : (at_1^2, 2at_1) and (at_2^2 , 2at_2)

Find the distance between the points : ( at_1^2 , 2at_1) and (at_2 ^2, 2at_2)

The normala at any point P (t^2, 2t) on the parabola y^2 = 4x meets the curve again at Q . Prove that the area of Delta AQP is k/|t| (1+t^2) (2+t^2) .

If AB and CD are two focal chords of the parabola y^2 = 4ax , then the locus of point of intersection of chords AC and BD is the directrix of the parabola. Statement 2: If (at^2_1, 2at_1) and (at^2_2, 2at_2) are the ends of a focal chord of the parabola y^2 = 4ax , then t_1 t_2 = -1 .

Find the area of the triangle whose vertices are : (at_1^2, 2at_1), (at_2^2 , 2at_2), (at_3^2, 2at_3)

(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 .

Let the tangents to the parabola y^(2)=4ax drawn from point P have slope m_(1) and m_(2) . If m_(1)m_(2)=2 , then the locus of point P is

If the normal to the parabola y^2=4a x at point t_1 cuts the parabola again at point t_2 , then prove that t2 2geq8.