If `t_1,t_2` and `t_3` are distinct, the points `(t_1, 2at_1 + at_1^3), (t_2, 2at_2 + at_2^3), (t_3, 2at_3 +at_3^3)` are collinear then show that `t_1+t_2+t_3=0`.

If t_1, t_2 and t_3 are distinct, the points (t_1, 2at_1+at_1^3), (t_2,2at_2+at_2^3) and (t_3,2at_3+at_3^3) are collinear if

Find the locus of the point (t^2-t+1,t^2+t+1),t in R .

Find the locus of the point (t^2+t+1,t^2-t+1),t in Rdot

Using determinant : find the area of the triangle whose vertices are (at_1^2,2at_1),(at_2^2,2at_2) and (at_3^2,2at_3)

If the Verticesof a triangle are P(at_1,a/t_1) , Q(at_2,a/t_2) and R(at_3,a/t_3) Show that the orthocentre of the triagle lies on xy = a^2 .

The vertices of a triangle are [a t_1t_2,a(t_1 +t_2)] , [a t_2t_3,a(t_2 +t_3)] , [a t_3t_1,a(t_3 +t_1)] Then the orthocenter of the triangle is (a) (-a, a(t_1+t_2+t_3)-at_1t_2t_3) (b) (-a, a(t_1+t_2+t_3)+at_1t_2t_3) (c) (a, a(t_1+t_2+t_3)+at_1t_2t_3) (d) (a, a(t_1+t_2+t_3)-at_1t_2t_3)

The vertices of a triangle are [at_1t_2, a(t_1+t_2)],[at_2t_3,a(t_2+t_3)],[at_3t_1,a(t_3+t_1)] Find the orthocentre of the triangle.

If the points P (0,3,2), Q(1,2,-2) and R (4,-1,t) are collinear, find the value of t.

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 .

Let C be incircle of A B Cdot If the tangents of lengths t_1,t_2 and t_3 are drawn inside the given triangle parallel to sidese a , b and c , respectively, the (t_1)/a+(t_2)/b+(t_3)/c is equal to 0 (b) 1 (c) 2 (d) 3