Question

If the points \[A{\text{ }}\left( {5,{\text{ }}a,{\text{ }} -1} \right)\], \[{\text{B }}\left( {2,{\text{ - 7}},{\text{ }}k} \right)\] and \[{\text{P }}\left( {\dfrac{{17}}{4},{\text{ }}\dfrac{{11}}{4},{\text{ }}0} \right)\] are collinear, then the ratio in which P divides AB is
A) 1:2
B) 3:1
C) 2:1
D) 1:3

Answer 1

Hint: The line segment AB will be divided by P in a certain ratio. This ratio can be calculated using the section formula. As these points are collinear, they all will lie on the same line. The coordinates of the points are given and thus can be substituted in the formula so as to find the required ratio.
Formula to be used:
$M\left( {x,y,z} \right) = \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},{\text{ }}\dfrac{{m{y_2} + n{y_1}}}{{m + n}},{\text{ }}\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$
M is the point that divides the line segment, x, y and z represents the coordinates of the three respective points.

Complete step-by-step answer:
It is given that the three points are collinear i.e. they lie on the same line. Point P divides these line segments, the two divided parts can and cannot be equal. We can calculate the ratio for this division by using the section formula which is given as:
If the points P and Q have coordinates $\left( {{x_1},{y_1},{z_1}} \right)$ and $\left( {{x_2},{y_2},{z_2}} \right)$ respectively. M be the line segment dividing this line in the ratio m:n, then according to the sectional formula:
$M\left( {x,y,z} \right) = \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},{\text{ }}\dfrac{{m{y_2} + n{y_1}}}{{m + n}},{\text{ }}\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$
The given points of line segment AB are \[A{\text{ }}\left( {5,{\text{ }}a,{\text{ }} -1} \right)\] and \[{\text{B }}\left( {2,{\text{ - 7}},{\text{ }}k} \right)\]. Let point \[{\text{P }}\left( {\dfrac{{17}}{4},{\text{ }}\dfrac{{11}}{4},{\text{ }}0} \right)\] divides this line segment in ratio m:n. then by applying sectional formula:
$M\left( {x,y,z} \right) = \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},{\text{ }}\dfrac{{m{y_2} + n{y_1}}}{{m + n}},{\text{ }}\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$
Here,
\[M\left( {x,y,z} \right) = {\text{P }}\left( {\dfrac{{17}}{4},{\text{ }}\dfrac{{11}}{4},{\text{ }}0} \right)\]
\[
  {x_1} = 5 \\
  {x_2} = 2 \\
 \]
$
 {y_1} = a \\
 {y_2} = - 7 \\
 $
$
  {z_1} = - 1 \\
  {z_2} = k \;
 $
Substituting the values, we get:
$\left( {\dfrac{{17}}{4},{\text{ }}\dfrac{{11}}{4},{\text{ }}0} \right) = \left( {\dfrac{{2m + 5n}}{{m + n}},{\text{ }}\dfrac{{ - 7m + an}}{{m + n}},{\text{ }}\dfrac{{km - 1n}}{{m + n}}} \right)$
We can find the value of m and n by equating either coordinate on both sides of the equations. But, one of the coordinates of both y and z are constants (a and k). so, we will equate the coordinates of x on both sides of the equation to find the required value of m and n.
\[ \Rightarrow \left( {\dfrac{{2m + 5n}}{{m + n}}} \right) = \dfrac{{17}}{4}\]
By cross multiplication, we get:
$
   \Rightarrow 4\left( {2m + 5n} \right) = 17\left( {m + n} \right) \\
   \Rightarrow 8m + 20n = 17m + 17n \\
   \Rightarrow 8m - 17m = 17n - 20n \\
   \Rightarrow - 6m = - 3n \\
   \Rightarrow \dfrac{m}{n} = \dfrac{{ - 3}}{{ - 9}} \\
   \Rightarrow \dfrac{m}{n} = \dfrac{1}{2} \\
   \Rightarrow m:n = 1:3 \;
 $
Therefore, the ratio in which P divides AB is $1:3$ and the correct option is D).
So, the correct answer is “Option D”.

Note: As we can write the ratio into fraction as $a:b = \dfrac{a}{b}$, thus the fraction can be written in the ratio as per the same relationship. The line segment can divide the line internally or externally, if it is not mentioned, it is taken to be as internal division only. For external division, all the positive signs in the sectional formula changes to the negative. If the line is divided equally and the point dividing is the point then the ratio of m and n is 1:2