FIBRE TO FABRIC L1

Two planes P_1 and P_2 pass through origin. Two lines L_1 and L_2 also passingthrough origin are such that L_1 lies on P_1 but not on P_2, L_2 lies on P_2 but not on P_1 A,B, C are there points other than origin, then prove that the permutation [A', B', C'] of [A, B, C] exists. Such that: (a) A lies on L1, B lies on P1 not on L1, C does not lie on P1 . (b) A lies on L2, B lies on P2 not on L2, C' does not lies on P2.

ABCD is a square of length a, a in N , a > 1. Let L_1, L_2 , L_3... be points on BC such that BL_1 = L_1 L_2 = L_2 L_3 = .... 1 and M_1,M_2 , M_3,.... be points on CD such that CM_1 = M_1M_2= M_2 M_3=... = 1 . Then sum_(n = 1)^(a-1) ((AL_n)^2 + (L_n M_n)^2) is equal to :

Consider the lines given by L_1: x+3y-5=0 L_2:3x-k y-1=0 L_3:5x+2y-12=0 Column I|Column II L_1,L_2,L_3 are concurrent if|p. k=-9 One of L_1,L_2,L_3 is parallel to at least one of the other two if|q. k=-6/5 L_1,L_2,L_3 form a triangle if|r. k=5/6 L_1,L_2,L_3 do not form a triangle if|s. k=5

An optical fibre (RI = 1.72) is surrounded by a glass coating (RI= 1.50). Find the critical angle for total internal reflection at the fibre-glass interface.

If L_1&L_2 are the lengths of the segments of any focal chord of the parabola y^2=x , then (a) 1/(L_1)+1/(L_2)=2 (b) 1/(L_1)+1/(L_2)=1/2 (c) 1/(L_1)+1/(L_2)=4 (d) 1/(L_1)+1/(L_2)=1/4

theta_1 and theta_2 are the inclination of lines L_1 and L_2 with the x-axis. If L_1 and L_2 pass through P(x_1,y_1) , then the equation of one of the angle bisector of these lines is

If l, l_(1), l_(2) and l _(3) be respectively the radii of the in-circle and the three escribed circles of a Delta ABC, then find l_(1)l_(2) l_(3)-l(l_(1)l_(2) +l_(2) l_(3)+l_(1) l_(3)).

Given equation of line L_1 is y = 4 Write the slope of line L_1 if L_2 is the bisector of angle O.

If L_1 = 2.02 m pm 0.01 m , L_2 = 1.02 m pm 0.01 m , determine L_1 + 2L_2

Consider three converging lenses L_1, L_2 and L_3 having identical geometrical construction. The index of refraction of L_1 and L_2 are mu_1 and mu_2 respectively. The upper half of the lens L_3 has a refractive index mu_1 and the lower half has mu_2 . A point object O is imaged at O_1 by the lens L_1 and at O_2 by the lens L_2 placed in same position . If L_3 is placed at the same place.