Find the equation of the plane containing the lineL1 , and parallel to the lineL

Question

Find the equation of the plane containing the lineL1 , and parallel to the lineL2 , where:

L1 : (x,y,z) = (1,2,-2) + t (-2,0,-2).

L2 : is the intersection of the planes1x1y3z=2 and3x3y2z=1

Write your answer in the form 11x+By+Cz=D, and givethe values of B, C and D as your answer.
B=? C=? D=? Find the equation of the plane containing the lineL1 , and parallel to the lineL2 , where:

L1 : (x,y,z) = (1,2,-2) + t (-2,0,-2).

L2 : is the intersection of the planes1x1y3z=2 and3x3y2z=1

Write your answer in the form 11x+By+Cz=D, and givethe values of B, C and D as your answer.
B=? C=? D=?

Explanation / Answer

QuestionDetails: Find the equation of the plane containing the lineL1 , and parallel to the lineL2 , where:
L1 : (x,y,z) = (1,2,-2) + t (-2,0,-2).
L2 : is the intersection of the planes1x1y3z=2 and3x3y2z=1
THAT IS ,L2 : is the intersection of the planes ,1x1y3z+2=0 AND3x+3y+2z1=0
DRS OF THESE 2 PLANES ARE [1,-1,-3] ; [3,3,2]
LET THE DRS OF L2 BE A,B,C.
L2 : is the intersection of the planes ,1x1y3z+2=0 AND3x+3y+2z1=0 AND HENCE
L2 LIES ON THESE 2 PLANES AND SO IS PERPENDICULAR TO NORMALS
TO THE 2 PLANES. HENCE WE GET
1A-1B-3C=0.....................................................1
3A+3B+2C=0....................................................2
EQN.2 - 3 *EQN.1 GIVES
6B+11C=0..................B=-11C/6
A=B+3C=-11C/6 + 3 = 7C/6
HENCE DRS OF L2 ARE ARE 7C/6 ; -11C/6 ; C...OR....(7,-11,6).....................3
REQD.PLANE IS PARALLEL TO THIS LINE . HENCE ITS NORMAL IS
PERPENDICULAR TO THE LINE
LET THE DRS OF REQD. PLANE BE L,M,N.
SO WE GET
7L-11M+6N =0.....................................................................4

REQUIRED PLANE CONTAINS............L1 : (x,y,z) = (1,2,-2) + t(-2,0,-2).
PUTTING T=0, WE GET ..(1,2,2) IS A POINT ON L1
PUTTING T=-1, WE GET ..(3,2,0) IS A POINT ON L1
HENCE (1,2,2) AND (3,2,0) ARE ON THE REQD. PLANE.....................................4
EQN. OF PLANE WILL BE OF THE TYPE
L(X-3)+M(Y-2)+N(Z-0)=0...........................5
POINT (1,2,2) LIES ON THE PLANE .HENCE
L(1-3)+M(2-2)+N(2-0)=0........OR.........
-2L+2N=0......L=N...............6
PUTTING IN EQN.4
7N-11M+6N =0...................13N=11M............................7
TAKING M=13 , WE GET N=11=L.....PUTTING IN EW.5
11(X-3)+13(Y-2)+11(Z-0)=0...........................OR
11X+13Y+11Z = 59
Write your answer in the form 11x+By+Cz=D, and givethe values of B, C and D as your answer.
B=?12 C=?11 D=?59
CHECK THE ARITHMETIC ALWAYS FOLLOWING THE METHOD

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