The chord `A B` of the parabola `y^2=4a x` cuts the axis of the parabola at `Cdot` If `A-=(a t_1^ 2,2a t_1),B-=(a t_2^2,2a t_2)` , and `A C : A B=1:3,` then prove that `t_2+2t_1=0` .

The points (a t_1^2, 2 a t_1),(a t_2^2, 2a t_2) and (a ,0) will be collinear, if

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 (t_2)^2geq8.

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.

Statement I: The lines from the vertex to the two extremities of a focal chord of the parabola y^2=4ax are perpendicular to each other. Statement II: If the extremities of focal chord of a parabola are (at_1^2,2at_1) and (at_2^2,2at_2) , then t_1t_2=-1 .

If the normal at (1,2) on the parabola y^(2)=4x meets the parabola again at the point (t^(2),2t) then the value of t is

If the normal at(1, 2) on the parabola y^(2)=4x meets the parabola again at the point (t^(2), 2t) then the value of t, is

The chord AC of the parabola y^(2)=4ax subtends an angle of 90^(@) at points B and D on the parabola. If points A, B, C and D are represented by (at_(i)^(2), 2at_(i)), i=1,2,3,4 respectively, then find the value of |(t_(2)+t_(4))/(t_(1)+t_(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

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 .

If t_1a n dt_2 are the ends of a focal chord of the parabola y^2=4a x , then prove that the roots of the equation t_1x^2+a x+t_2=0 are real.