Show that the equation by `by+cz+d=0` represents a plane parallel to the x-axis. Find the equation of a plane which is parallel to the x-axis and passes through the points A(2,3,1) and B(4,-5,3).

Explanation

The given equation is 0.x+by+cz+d=0 which is of the form ax + by +cz+d=0
Hence the given equation represent a plane
Dr's of the normal to this plane are 0,bc
D.r 's of the x axis are 1,0 ,0
Now `0xx1+bxx0+cxx0=0`
This shows that the given plane is paralle to hte x axis
Thus the equaiton of a plane parallel to hte x axis is
by+cz+d=0
If it passing throught the points A (2,3,1) and B (4,-5,3) we have
3b+c+d=0
-5b+3c+d=0
on solving (ii) and (iii) by cross multiplication we get
`(b)/(1-3)=(c )/(-5-3)=(d )/(9+5)`
`rarr (b)/(-2)=(c )/(-8)=(d)/(14) rarr (b)/(1)=(c )/(4)=(d)/(-7)=k` (say)
`rarr` b=k,c = 4k and d=-7k
putting these values in (i) we get
`ky +4kz-7k =0 rarr y+4z-7=0`
Hence the required equation of the plane is y + 4z-7 =0