Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

The point which divides the line joining the points $\left( {1,3,4} \right)$ and $\left( {4,3,1} \right)$ internally in the ratio$2:1$ is
$
  a.\left( {2, - 3,3} \right) \\
  b.\left( {2,3,3} \right) \\
  c.\left( {\dfrac{5}{2},3,\dfrac{5}{2}} \right) \\
  d.\left( { - 3,3,2} \right) \\
  e.\left( {3,3,2} \right) \\
$

seo-qna
Last updated date: 19th Jul 2024
Total views: 452.1k
Views today: 10.52k
Answer
VerifiedVerified
452.1k+ views
Hint: Apply section formula if a line divide the points $\left( {{x_1},{y_1},{z_1}} \right)$ and $\left( {{x_2},{y_2},{z_2}} \right)$ in the ratio $\left( {m:n} \right)$ internally then
$x = \dfrac{{\left( {m.{x_2} + n.{x_1}} \right)}}{{m + n}},y = \dfrac{{\left( {m.{y_2} + n.{y_1}} \right)}}{{m + n}},z = \dfrac{{\left( {m.{y_2} + n.{y_1}} \right)}}{{m + n}}$

seo images


Given points are $\left( {1,3,4} \right)$ and $\left( {4,3,1} \right)$.
Let $\left( {1,3,4} \right) \equiv \left( {{x_1},{y_1},{z_1}} \right)$
And $\left( {4,3,1} \right) \equiv \left( {{x_2},{y_2},{z_2}} \right)$
Now, let the point $\left( {x,y,z} \right)$ divide the line internally in the ratio $2:1$
So, by section formula if a line divide the points $\left( {{x_1},{y_1},{z_1}} \right)$ and $\left( {{x_2},{y_2},{z_2}} \right)$ in the ratio $\left( {m:n} \right)$ internally then
$x = \dfrac{{\left( {m.{x_2} + n.{x_1}} \right)}}{{m + n}},y = \dfrac{{\left( {m.{y_2} + n.{y_1}} \right)}}{{m + n}},z = \dfrac{{\left( {m.{y_2} + n.{y_1}} \right)}}{{m + n}}$
Here $m = 2{\text{ & }}n = 1$
$x = \dfrac{{\left( {2.\left( 4 \right) + 1.\left( 1 \right)} \right)}}{{2 + 1}},y = \dfrac{{\left( {2.\left( 3 \right) + 1.\left( 3 \right)} \right)}}{{2 + 1}},z = \dfrac{{\left( {2.\left( 1 \right) + 1.\left( 4 \right)} \right)}}{{2 + 1}}$
$ \Rightarrow \left( {x,y,z} \right) = \left( {\dfrac{9}{3},\dfrac{9}{3},\dfrac{6}{3}} \right) = \left( {3,3,2} \right)$

Hence option (e) is correct.

Note: In such types of questions the key concept we have to remember is that always recall the section formula which is stated above then substitute the given points in this formula and simplify, we will get the required points which divide the line internally in the ratio $2:1$.