Find the equation of the plane passing through the lineof intersection of the planes `x+2y+3z-5=0` and `3x-2y-z+1=0` and cutting off equal intercepts on the x-axis and z-axis.

Answer

`5x+2y+5z-9=0`

Explanation

Let the required plane be
`(x+2y+3z-5)+lambda(3x-2y-z+1)=0`.
Then, `(1+3lambda)x+(2-2lambda)y+(3-lambda)z-5+lambda=0`.
`(1+3lambda)x+(2-2lambda)y+(3-lambda)z=5-lambda`
`rArr (1+3lambda)/(5-lambda)x+((2-2lambda))/((5-lambda))y+((3-lambda))/((5-lambda))=1`.
`rArr x/((5-lambda)/(1+3lambda))+y/((5-lambda)/(2-2lambda))+z/((5-lambda)/(3-lambda))=1`.
Since the intercepts on the x-axis and z-axis are equal, we have
`(5-lambda)/(1+3lambda)=(5-lambda)/(3-lambda) rArr 3-lambda=1+3lambda rArr 4lambda=2 rArr lambda=1/2`.