#### Question

Find the equation of the plane which contains the line of intersection of the planes

`vecr.(hati-2hatj+3hatk)-4=0" and"`

`vecr.(-2hati+hatj+hatk)+5=0`

and whose intercept on x-axis is equal to that of on y-axis.

#### Solution

`vecr.(hati-2hatj+3hatk)-4=0`

`vecr.(-2hati+hatj+hatk)+5=0`

`vecr.(hati-2hatj+3hatk)+lambda{vecr.(-2hati+hatj+hatk)}-4+5lambda = 0`

`=>vecr.[(1-2lambda)hati+(-2+lambda)hatj+(3+lambda)hatk]-4+5lambda=0`

`=>(1-2lambda)x+(-2+lambda)y+(3+lambda)z=-5lambda+4`

`=>x/((-5lambda+4)/(1-2lambda))+y/((-5lambda+4)/(-2+lambda))+z/((-5lambda+4)/(3+lambda))=1`

`:.(-5lambda+4)/(1-2lambda) =(-5lambda+4)/(-2+lambda)`

⇒ 1- 2λ = -2 + λ

⇒ -3λ = -3

⇒ λ = 1

∴ Equation of the required plane

- x - y + 4z = -1

x + y - 4z - 1 = 0

Vector eq^{n} of the required Plane

`=>vecr.(hati+hatj-4hatk)-1=0`