Let `l_1a n dl_2` be the two skew lines. If `P ,Q` are two distinct points on `l_1n dR , S` are two distinct points on `l_2,` then prove that `P R` cannot be parallel to `Q Sdot`

Let l_(1) and l_(2) be the two skew lines. If P, Q are two distinct points on l_(1) and R, S are two distinct points on l_(2) , then prove that PR cannot be parallel to QS.

Let L_1:3x+4y=1a n dL_2=5x-12 y+2=0 be two given lines. Let image of every point on L_1 with respect to line L lies an L_2 then possible equations of L can be (a) 14 x+112 y-23=0 (b) 64 x-8y-3=0 (c) 11 x-4y=0 (d) 52 y-45 x=7

In Figure, common tangents P Qa n dR s to two circle intersect at Adot Prove that P Q=R Sdot

Let L_1 and L_2 be two lines intersecting at P If A_1,B_1,C_1 are points on L_1,A_2, B_2,C_2,D_2,E_2 are points on L_2 and if none of these coincides with P, then the number of triangles formed by these 8 points is

P is a point equidistant from two lines l and m intersecting at a point A , Show that A P bisects the angle between them.

Let L_1=0a n dL_2=0 be two fixed lines. A variable line is drawn through the origin to cut the two lines at R and SdotPdot is a point on the line A B such that ((m+n))/(O P)=m/(O R)+n/(O S)dot Show that the locus of P is a straight line passing through the point of intersection of the given lines R , S , R are on the same side of O)dot

If P is any point on the plane l x+m y+n z=pa n dQ is a point on the line O P such that O P.O Q=p^2 , then find the locus of the point Qdot

If P is any point on the plane l x+m y+n z=pa n dQ is a point on the line O P such that O PdotO Q=p^2 , then find the locus of the point Qdot

Statement I: Two point coherent sources of light S_(1) and S_(2) are placed on a line as shown in figure. P and Q are two points on that line. If at point P maximum intensity is observed, then maximum intensity should also be observed at Q. Statement II: In figure the distance |S_(1) P - S_(2) P| is equal to distance |S_(1) Q - S_(2) Q| .

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.