Find the equation of a plane passes through the points `(0,0,0)` and `(1,3,5)` and parallel to the line `(x)/(-2) = y/1 = (z+3)/(4)`.

Explanation

Let equation of the plane passing through the point `(0,0,0)` is
`a(x-0)+b(y-0)+c(z-0) = 0`
`rArr ax+ by+cz = 0 "….."(1)`
This plane passes throough the point `(1,3,5)`.
`:. a+3b+5+5c=0"……"(2)`
This plane is parallel to the line `(x)/(-2) = (y)/(1) = (z+3)/(4)`.
`:. -2a+b+4c = 0 "....."(3)`
Solving eqs. (2) and (3) by cross multiplication method.
`(a)/(12-5) = (b)/(-10-4) = (c)/(1+6)`
`rArr a/7 = (b)/(-14) = c/1 = k`
`:. a = k, b = - 2 k, c = k`
From eq. (1)
`kx - 2ky + kz = 0`
`rArr x - 2y + z = 0`
Which is the requried equation of the plane.