Question

The line joining the points $( - 2,1, - 8)$ and $(a,b,c)$ is parallel to the line whose direction ratios are $6,2,3$ then find the value of $a,b,c$
A. $(4,3, - 5)$
B. $(1,2,\dfrac{{ - 13}}{2})$
C. $(10,5, - 2)$
D. $\left( { - 5,3,4} \right)$

Answer 1

Hint: Given, the line joining the points $( - 2,1, - 8)$ and $(a,b,c)$ is parallel to the line whose direction ratios are$6,2,3$. We have the value of $a,b,c$.First, we will find the direction ratios. Then we compare with given direction ratios using the concept that if lines are parallel then their direction ratios are equal.

Formula used: If two endpoints of a vector are given by $P(x,y,z)$ and $Q(a,b,c)$, then
the direction ratio of vector PQ\[ = \left( a-x, b-y, c-z \right)\]

Complete step by step solution:
Given, the line joining the points $( - 2,1, - 8)$ and $(a,b,c)$ is parallel to the line whose direction ratios are $6,2,3$.
If two point is given by $(x,y,z)$ and $(a,b,c)$, then
the direction ratio\[ = {\left( a-x, b-y, c-z \right)}\]
$ \Rightarrow (a + 2,b - 1,c + 8)$ are the direction ratio
We know that if lines are parallel then their direction ratios are equal
$ \Rightarrow (a + 2,b - 1,c + 8) = (6,2,3)$
After comparing
$a + 2 = 6$
Shifting 2 to another side
$a = 6 - 2$
After solving the above equation
$a = 4$
$b - 1 = 2$
Shifting 1 to other sides
$b = 2 + 1$
After solving the above equation
$b = 3$
$c + 8 = 3$
Shifting 8 to another side
$c = 3 - 8$
After solving the above equation
$c = - 5$
So, $(a,b,c) = (4,3, - 5)$

So, option (A) is the correct answer.

Note: Students should pay attention while solving the question to avoid any mistakes. They should use concepts properly to get the correct answer. They should use the concept correctly that if lines are parallel then their direction ratios are equal in order to get an error-free accurate required solution.