Question

Let P be a plane $lx + my + nz = 0$ containing the line $\frac{{1 - x}}{1} = \frac{{y + 4}}{2} = \frac{{z + 2}}{3}$. If plane P divides the segment AB joining points $A( - 3, - 6,1)$ and $B(2,4, - 3)$ in ratio $k:1$ then the value of $k$ is equal to:
a) $1.5$
b) $2$
c) $4$
d) $3$

Answer 1

Hint: Given P be a plane $lx + my + nz = 0$ containing the line $\frac{{1 - x}}{1} = \frac{{y + 4}}{2} = \frac{{z + 2}}{3}$. And plane P divides the segment AB joining points $A( - 3, - 6,1)$ and $B(2,4, - 3)$ in ratio $k:1$.
Firstly, we will find the equation of the plane using the equation of the given line then, we will find the value of $k$.

Complete step by step Solution:
In the question we are given the equation of line $\frac{{1 - x}}{1} = \frac{{y + 4}}{2} = \frac{{z + 2}}{3}$
The line lies on the plane
$ - l + 2m + 3n = 0$ (1)
Point $(1, - 4, - 2)$ lies on the plane
$l - 4m - 2n = 0$ (2)
 now we will add equation (1) and equation (2)
$ - l + 2m + 3n + l - 4m - 4n = 0$
after adding the equation l gets eliminated and the equation is formed in terms of m and n
$ - 2m + n = 0$
taking n on the other side
$2m = n$
hence the value of m in terms of n
$m = \frac{n}{2}$
now we will put this value of m in equation (1)
$ - l + n + 3n = 0$
and we get the value of l in terms of m
$ \Rightarrow l = 4m$
forming the ratio using all the equations formed above
$l:m:n::4n:\frac{n}{2}:n$
now we will simplify the equation by removing all the fractions from the equations
$ \Rightarrow l:m:n::8n:n:2n$
from the above equation, we will remove the term n from right-hand side ratio to make it simple
$ \Rightarrow l:m:n::8:1:2$
Hence, $8x + y + 2z = 0$ be the equation of the plane.
Given, that P divides AB in ratio $k:1$
$8(\frac{{ - 3 + 2k}}{{k + 1}}) + (\frac{{ - 6 + 4k}}{{k + 1}}) + 2(\frac{{1 - 3k}}{{k + 1}}) = 0$
By solving the above equation we get
$16k - 24 + 4k - 6 - 6k + 2 = 0$
taking all the similar terms together and constant at one side we get
$ \Rightarrow 14k = 28$
value of k by taking 14 on the other side
$k = 2$

Hence, the correct option is (b).

Note:Students mostly make mistakes while finding the equation of the plane. We must be careful while finding equations. And also concentrate while calculating the value of k to avoid calculation errors.