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' lies on `L_2`, B' lies on P2 not on `L_2`, C' does not lies on `P_2`.

Two planes P_1 and P_2 pass through origin. Two lines L_1 and L_2 also passing through 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 lies on L_1 , B lies on P_1 not on L_1 , C does not lie on P_1 .

Of all the line segments drawn from a point P to a line l not containing P, let PD be the shortest. If B and C are points on l such that D is the mid-point of BC, prove that PB = PC.

Fill ups A plane 'p' passes through the point P(1,-2,3). If O is the origin and OPbotp , then equation of the plane p is………….. .

If each of the points (x_1,4),(-2,y_1) lies on the line joining the points (2,-1)a n d(5,-3) , then the point P(x_1,y_1) lies on the line.

C_1 and C_2 are two concentric circles, the radius of C_2 being twice that of C_1 . From a point P on C_2 , tangents PA and PB are drawn to C_1 . Prove that the centroid of the triangle PAB lies on C_1 .

Given planes P_1:cy+bz=x P_2:az+cx=y P_3:bx+ay=z P_1, P_2 and P_3 pass through one line, if

A line L_1 passing through a point with position vector p=i+2h+3k and parallel a=i+2j+3k , Another line L_2 passing through a point with direction vector to b=3i+j+2k . Q. The minimum distance of origin from the plane passing through the point with position vector p and perpendicular to the line L_2 , is

In R^3 , let L be a straight line passing throughthe origin. Suppose that all points on L are at a constant distance from the two planes: P_1: x+2y-z+1=0 and P_2: 2x-y-z+1=0 . Let M be the locus of the feet of the perpendiculars drawn from the points on L t the plane P_1 . Which of the following point(s) lie(s) on M?

Find the equation of plane passing through the points : (2, - 1, 1), B (4, 3, 2) and C (6, 5, -2). Also, prove. that the point (5, -1,-2:) lies on the plane given by A, B and C.

Let C_1 and C_2 be parabolas x^2 = y - 1 and y^2 = x-1 respectively. Let P be any point on C_1 and Q be any point C_2 . Let P_1 and Q_1 be the reflection of P and Q, respectively w.r.t the line y = x then prove that P_1 lies on C_2 and Q_1 lies on C_1 and PQ >= [P P_1, Q Q_1] . Hence or otherwise , determine points P_0 and Q_0 on the parabolas C_1 and C_2 respectively such that P_0 Q_0 <= PQ for all pairs of points (P,Q) with P on C_1 and Q on C_2