If `t-1 and -t-1, t in R` are the roots of `(a+2)x^2 +2ax -1 =0`,then complete set of values of 'a' is

Tangents drawn to the parabola y^(2)=8x at the points P(t_(1)) and Q(t_(2)) intersect at a point T and normals at P and Q intersect at a point R such that t_(1) and t_(2) are the roots of equation t^(2)+at+2=0;|a|>2sqrt(2) then locus of R is

If r,s,t are the roots of the equation x(x-2)(3x-7)=2 ,then the value of tan^(-1)(r)+tan^(-1)(s)+tan^(-1)(t) is

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 points (a ,0),(a t1 ^ 2,\ 2a t_1)a n d\ a t2^ 2,\ 2a t_2) are collinear, write the value of t_1t_2dot

If t_(1) and t_(2) are the extremities of any focal chord of y^(2) = 4ax " then " t_(1)t_(2) is _____

If t_(1) and t_(2) are the ends of a focal chord of the parabola y^(2)=4ax, then prove that the roots of the equation t_(1)x^(2)+ax+t_(2)=0 are real.

If area of pentagon PQRST be 7 ,where P(-1,-1),Q(2,0),R(3,1),S(2,2) and T(-1,t),t>0, then the value of t is

Let f (x) = int _(0)^(x) ((a -1) (t ^(2)+t+1)^(2) -(a+1)(t^(4)+t ^(2) +1)) dt. Then the total numbr of integral values of 'a' for which f'(x)=0 has no rel roots is

Let f (x) = int _(0)^(x) ((a -1) (t ^(2)+t+1)^(2) -(a+1)(t^(4)+t ^(2) +1)) dt. Then the total number of integral values of 'a' for which f'(x)=0 has no real roots is