A plane meets the coordinate axes in `A ,B ,C` such that eh centroid of triangle `A B C` is the point `(p ,q ,r)dot` Show that the equation of the plane is `x/p+y/q+z/r=3.`

Explanation

Let the required equation of the plane be `x/a+y/b+z/c=1`……………(i)
Then, clearly the plane meets the coordinates axis at `A(a,0,0),B(0,b,0)` and C(0,0,c).
Since, the centroid of `triangleABC` isG(p,q,r), we have
`(a+0+0)/(3)=p, (0+b+0)/(3)` and `(0+0+c)/3=r`
`rArr a=3p, b=3q` and `c=3r`.
Putting, these values of a,b,c in (i), we get
`x/(3p)+y/(3q)+z/(3r)=1 rArr x/p+y/q+z/r=3`.
Hence, the required equation of the plane is `x/p +y/q+z/r=3`.