If `A(at^2, 2at), B((a)/(t^2) , - (2a)/(t))` and `C(a, 0)` be any three points, show that `1/(AC) + 1/(BC)` is independent of `t`.

If P and Q are two points whose coordinates are (a t^2,2a t)a n d(a/(t^2),(2a)/t) respectively and S is the point (a,0). Show that 1/(S P)+1/(s Q) is independent of t.

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 points (2,0)(3,2)(5,4) and (0,t) are concylic, show that t=2 or 17.

If x=(2t)/(1+t^2) , y=(1-t^2)/(1+t^2) , then find (dy)/(dx) .

If x=(1-t^(2))/(1+t^(2)) and y=(2t)/(1+t^(2)) , then (dy)/(dx) is equal to

If x=(1-t^2)/(1+t^2) and y=(2t)/(1+t^2) ,then (dy)/(dx)=

If x=a t^2,y=2a t , then (d^2y)/(dx^2) is equal to (a) -1/(t^2) (b) 1/(2a t^2) (c) -1/(t^3) (d) 1/(2a t^3)

A=(sqrt(1-t^2)+t, 0) and B=(sqrt(1-t^2)-t, 2t) are two variable points then the locus of mid-point of AB is

Locus of the middle points of the line segment joining P(0, sqrt(1-t^2) + t) and Q(2t, sqrt(1-t^2) - t) cuts an intercept of length a on the line x+y=1 , then a = (A) 1/sqrt(2) (B) sqrt(2) (C) 2 (D) none of these

If a circle intersects the parabola y^(2) = 4ax at points A(at_(1)^(2), 2at_(1)), B(at_(2)^(2), 2at_(2)), C(at_(3)^(2), 2at_(3)), D(at_(4)^(2), 2at_(4)), then t_(1) + t_(2) + t_(3) + t_(4) is