If all three planes `P_1: (a+1)^(3)x+(a+2)^(3)y+(a+3)^(3)z=0` ,`P_2:(a+1)x+(a+2)y+(a+3)z=0` , `P_3: x+y+z=0` passes through a point other than origin. then `a` equals

The equations of three planes are P_(1):x-2y+z-3=0, P_(2):5x-y-z-8=0 and P_(3):x+y-z-7=0 , then the common solution of the three planes is.

(i) Find the equation of the plane passing through (1,-1,2) and perpendicular to the planes : 2 x + 3y - 2z = 5 , x + 2y - 3z = 8 . (ii) find the equation of the plane passing through the point (1,1,-1) and perpendicular to each of the planes : x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0 . (iii) Find the equation of the plane passing through the point (-1,-1,2) and perpendicular to the planes : 3x + 2y - 3z = 1 and 5x - 4y + z = 5.

Let P_(1) : 2x + y + z + 1 = 0 P_(2) : 2x - y + z + 3 = 0 and P_(3) : 2x + 3y + z + 5 = 0 be three planes, then the distance of the line of intersection of planes P_(1) = 0 and P_(2) = 0 from the plane P_(3) = 0 is

Let the equations of two planes be P_1: 2x-y+z=2 and P_2: x+2y-z=3 Equation of the plane which passes through the point (-1,3,2) and is perpendicular to each of the plane P_1 and P_2 is (A) x-3y-5z+20=0 (B) x+3y+5z-18=0 (C) x-3y-5z=0 (D) x+3y-5z=0

If |(x, x^2, x^3 +1), (y, y^2, y^3+1), (z, z^2, z^3+1)| = 0 and x ,y and z are not equal to any other, prove that, xyz = -1

If the line (x-2)/(3)=(y+1)/(2)=(z-1)/(2) intersects the plane 2x+3y-z+13=0 at a point P and plane 3x+y+4z=16 at a point Q then PQ is equal to

Consider the planes p _(1) : 2x + y+ z+4=0 p _(2) :y -z+4 =0 and p _(3) : 3x+ 2y +z+8=0 Let L_(1),I _(2), I_(3) be the lines of intersection of the planes P_(2)and P_(3), P_(3) and P _(1),P_(1) and P _(2) respectively. Then: