Let `w=zbarz +k_1z+k_2iz+lambda(1+i),k_1,k_2 in RR`. Let `Re(w)=0` be the circle C radius `1` in the first quadrant touching the line `y=1` and the y-axis . If the cuerve `Im(w)=0` intersects C at A and B, then `30(AB)^2` is equal to ____.

Let A be the area of the region in the first quadrant bounded by the x-axis the line 2y = x and the ellipse (x^(2))/(9) + (y^(2))/(1) = 1 . Let A' be the area of the region in the first quadrant bounded by the y-axis the line y = kx and the ellipse . The value of 9k such that A and A' are equal is _____

If the straight lines (x-1)/k = (y-2)/2 =(z-3)/3 and (x-2)/3 = (y-3)/k = (z-1)/2 intersect at a point, then integer k is equal to

Let the line L : sqrt(2)x y =alpha pass through the point of the intersection P (in the first quadrant) of the circle x^2 y^2 = 3 and the parabola x^2 = 2y . Let the line L touch two circles C_1 and C_2 of equal radius 2 sqrt 3 . If the centres Q_1 and Q_2 of the circles C_1 and C_2 lie on the y-axis, then the square of the area of the triangle PQ_1Q_2 is equal to ___________.

Let the point at which the circle passing through (0, 0) and (1, 0) touches the circle x^(2)+y^(2)=9 is P(h, k), then |k| is equal to

If the straight lines (x-1)/(k)=(y-2)/(2)=(z-3)/(3) and (x-2)/(3)=(y-3)/(k)=(z-1)/(2) intersect at a point, then the integer k is equal to

If a complex number z lie on a circle of radius (1)/(2) units, then the complex number omega=-1 +4z will always lie on a circle of radius k units, where k is equal to

Let v and w be distinct, randomly chosen roots (real or complex) of the equation z^(9)-1=0 .The probability that 1<=|v+w| is 1/k, then "k" is equal to