If `t_1!=t_2!=t_3` are the lines `t_1x+y=2at_1+at_1^3, t_2x+y=2at_2+at_2^3, t_3x+y=2at_3+at_3^3` are concurrent then `t_1+t_2+t_3=`

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

From a point P (h, k), in general, three normals can be drawn to the parabola y^2= 4ax. If t_1, t_2,t_3 are the parameters associated with the feet of these normals, then t_1, t_2, t_3 are the roots of theequation at at^2+(2a-h)t-k=0. Moreover, from the line x = - a, two perpendicular tangents canbe drawn to the parabola. If the tangents at the feet Q(at_1^2, 2at_1) and R(at_1^2, 2at_2) to the parabola meet on the line x = -a, then t_1, t_2 are the roots of the equation

If the line joining the points (at_1^2,2at_1)(at_2^2, 2at_2) is parallel to y=x then t_1+t_2=

If the line (x-1)/2=(y-2)/3=t intersects the line x+y=8 then t=

Prove that area of the trianlge whose vertices are (at_1^2,2at_1) , (at_2^2,2at_2),(at_3^2,2at_3) is a^2(t_1-t_2)(t_2-t_3)(t_3-t_1)

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)