Equation of plane passing through the line of intersection of given plane is
\(\vec r .(\hat i + \hat j + \hat k) + \lambda (2\hat i + 3\hat j - \hat k)= 10 -4 \lambda\)
where \(\vec r = x \hat i + y\hat j +z \hat k\)
⇒ \((x \hat i + y\hat j + z\hat k).((1 + 2\lambda)\hat i + (1 + 3\lambda)\hat j + (1 - \lambda)\hat k) = 10 - 4\lambda\) .....(i)
It is passing through point (-2, 3, 1).
\(\therefore -2 ( 1 + 2\lambda )+ 3(1+ 3\lambda)+ 1(1 - \lambda) = 10 - 4\lambda\)
⇒ \(- 2-4 \lambda + 3 + 9\lambda + 1 - \lambda = 10 - 4 \lambda\)
⇒ \(8 \lambda = 10 -2\)
⇒ \(8 \lambda = 8\)
⇒ \(\lambda = 1\)
Hence, the equation of required plane is
\(\vec r .(3 \hat i + 4 \hat j) = 6\) (From(i))
or \(3x + 4y = 6\)
