Three Dimensional Geometry

Line L will be parallel to the line of intersection of P1‌and‌P2
Let a,band c be the direction ratios of line L
⇒a+2b−c=0‌and‌2a−b+c=0
⇒a:b:c::1:−3:−5
Equation of line L is

x−0

1

=

y−0

−3

=

z−0

−5

Again foot of perpendicular from origin to plane P1 is (−

1

6

,−

1

3

,

1

6

)
∴Equation of projection of line L on plane P1 is

x+ 1 6

1

=

y+ 2 6

−3

=

z− 1 6

−5

=k
Clearly points 0,−

5

6

,−

2

3

‌and‌(−

1

6

,−

1

3

,

1

6

) satisfy the line of projection i.e,M.
Alternative Solution
Direction ratio of plane can be found by (

→

n1

×

→

n2

)×

→

n1

≡(13,−4,5)
So,equation of plane is 13x-4y+5z=0 and point (0,−

5

6

,−

2

3

)&(−

1

6

,−

1

3

,

1

6

) satisfy