Find two different planes whose intersection is the line x

mronjo7n

mronjo7n

Answered question

2021-11-13

Find the intersection of two separate planes that forms the line x = 1 + t, y = 2 - t, z = 3 + 2t. Write equations for each plane in the form Ax + By + Cz = D.

Answer & Explanation

Mollicchiuk

Mollicchiuk

Beginner2021-11-14Added 13 answers

L1=x+t, y=2t, z=3+2t
Let
P1:A1x+B1y+C1z=D1
P2:A2x+B2y+C2z=D2
The constants in the two planes equation determine if the two planes intersect with the line or not, hence we will choose two points, on the line, and then make them on the two planes as follow.
By putting t=0 in the line equation, we get the point P1=(1,2,3)
By putting t=1 in the line equation, we get the point P2=(2,1,5)
A1+2B1+3C1=D1
A2+2B2+3C2=D2
2A1+B1+5C1=D1
2A2+B2+5C2=D2
The two equations (1), (3) are two equations in four vars, hence we will choose two of them (A1=1 and B1=1) and then will solve the two equations for C1 and D1 as follows
5+3C1=D1
4+5C1=D1
C2=1
D2=10
Hence by using the values of constants we got in the planes equations, we have
P1:x+2y+12z=132
P2:x+3y+z=10
By multiplying the first equations by 2, we have
P1:2x+4y+z=13
P2:x+3y+z=10

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