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

I=int sqrt(1-t^(2))dt

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

If the normals at points t_1 and t_2 meet on the parabola, then (a) t_1t_2=1 (b) t_2=-t_1-2/(t_1) (c) t_1t_2=2 (d) none of these

P is a point on the line y+2x=1, and Q and R are two points on the line 3y+6x=6 such that triangle P Q R is an equilateral triangle. The length of the side of the triangle is (a) 2/(sqrt(5)) (b) 3/(sqrt(5)) (c) 4/(sqrt(5)) (d) none of these

The image of the centre of the circle x^2 + y^2 = 2a^2 with respect to the line x + y = 1 is : (A) (sqrt(2), sqrt(2) (B) (1/sqrt(2) , sqrt(2) ) (C) (sqrt(2), 1/sqrt(2)) (D) none of these

If x^2+y^2=t-1/t and x^4+y^4=t^2+1/t^2, then x^3y (dy)/(dx)= (a) 0 (b) 1 (c) -1 (d) none of these

If tan^(-1)(a+x)/a+tan^(-1)(a-x)/a=pi/6,t h e nx^2= 2sqrt(3)a (b) sqrt(3)a (c) 2sqrt(3)a^2 (d) none of these

If y=|cosx|+|sinx|,t h e n(dy)/(dx)a tx=(2pi)/3 is (1-sqrt(3))/2 (b) 0 (c) 1/2(sqrt(3)-1) (d) none of these

The direction cosines of line joining (1,-1,1) and (-1,1,1) are (A) 2,-2,0 (B) 1,-1,0 (C) 1/sqrt(2)-1/sqrt(2),0 (D) none of these

The locus of the point of intersection of the lines sqrt3 x- y-4sqrt3 t= 0 & sqrt3 tx +ty-4 sqrt3=0 (where t is a parameter) is a hyperbola whose eccentricity is: (a) sqrt3 (b) 2 (c) 2/sqrt3 (d) 4/3