If the point (a,0) lies on the line containing the point `(at_1^2 , 2at_1)` and `at_2^2,2at_2)`. Prove that `t_1t_2=-1`.

Find the equation of the line containing the points (1,2), (3,8)

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 .

Show that the points (at_(1)^(2),2at_(1)) , (at_(2)^(2),2at_(2)) and (a,0) are collinear if t_1t_2 =-1 .

If t_1 a n d t_2 are roots of eth equation t^2+lambdat+1=0, where lambda is an arbitrary constant. Then prove that the line joining the points (a t1^2,2a t_1)a d n(a t2^2,2a t_2) always passes through a fixed point. Also, find the point.

If each of the points (x_1,4),(-2,y_1) lies on the line joining the points (2,-1)a n d(5,-3) , then the point P(x_1,y_1) lies on the line.

Find the distance of the point (1,2,3) from the line joining the points (-1,2,5) and (2,3,4).

Find the vector equation of the line joining the points A(1,-1,2) and B(1,-2,0).

The points (a+1,1), (2a+1,3) and (2a+2,2a) are collinear if

Find the distance between the points (at_(1)^(2), 2 at_(1)) and (at_(2)^(2), 2 at_(2)) , where t_(1) and t_(2) are the roots of the equation x^(2)-2sqrt(3)x+2=0 and a gt 0 .