Question

STATEMENT 1:A point A $(1,0,7)$ is the mirror image of point B $(1,6,3)$ in the line$\dfrac{x}{1} = \dfrac{{y - 1}}{2} = \dfrac{{z - 2}}{3}$
STATEMENT 2: The line \[\dfrac{x}{1} = \dfrac{{y - 1}}{2} = \dfrac{{z - 2}}{3}\]bisects the line segment joining A $(1,0,7)$ and B $(1,6,3)$
A) Statement 1 is true statement, statement 2 is true. Statement 2 is a correct explanation of Statement 1
B) Statement 1 is true statement, statement 2 is true. Statement 2 is not correct explanation of Statement 1
C) Statement 1 is true statement, statement 2 is false.
D) Statement 1 is a false statement, statement 2 is true.

Answer 1

Hint:At first find out the midpoint of line segment AB. Then check whether it lies on the given line or not. This will check statement 2. Now check whether AB is perpendicular to the given line, which will check statement 1.

Complete step-by-step answer:
Let us first check statement 2. For the statement to be true we need to prove that the midpoint of AB lies on the given line.
Let line L $ = \dfrac{x}{1} = \dfrac{{y - 1}}{2} = \dfrac{{z - 2}}{3}$
Now let us find the midpoint of AB
Midpoint of AB$ = \dfrac{{coordinates\,of\,A + coordinates\,of\,B}}{2}$
Given A $(1,0,7)$ and B $(1,6,3)$
Let midpoint of AB be M which has coordinates:
$
   \equiv \left( {\dfrac{{1 + 1}}{2},\dfrac{{0 + 6}}{2},\dfrac{{7 + 3}}{2}} \right) \\
   \equiv \left( {1,3,5} \right) \\
 $
By substituting coordinates of line M in line L. We will check whether M lies on L or not.
Hence using $x = 1,y = 3,z = 5$in line L
$
   \Rightarrow \dfrac{1}{1} = \dfrac{{3 - 1}}{2} = \dfrac{{5 - 2}}{2} \\
   \Rightarrow 1 = 1 = 1 \\
 $
Since the condition is true
So line AB bisects L
So Statement-2 is true
Now we will check Statement-1
For it to be true we need to prove that line L is a perpendicular bisector of AB as statement says that point A is a mirror image of point B in line L.
We have already proved that line L bisects AB in statement 2.
Now we need to prove that AB is perpendicular to line L.
For that we will use their direction ratios and if ${a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0$, they are perpendicular.
The above condition is for two lines to be perpendicular, hence we will find the direction ratios.
Direction ratios of L$ = (1,2,3)$
Direction ratios of AB$ = $Coordinate of A $ - $ Coordinates of B
$
   = (1 - 1,6 - 0,3 - 7) \\
   = (0,6, - 4) \\
 $
Now we have,
$
  {a_1} = 1,{b_1} = 2,{c_1} = 3 \\
  {a_2} = 0,{b_2} = 6,{c_2} = - 4 \\
 $
${a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0 + 12 - 12 = 0$
Hence AB and L are perpendicular
Statement 1 is also true
But statement 2 is not the correct explanation of statement 1 as AB bisected by L does not mean L is perpendicular to AB.

So, the correct answer is “Option B”.

Note:In such questions you really need to take care of the statements given. You carefully need to understand them and then solve accordingly. Another difficult part is whether statement 2 is the correct explanation or not. You can check it by asking a question to yourself “does statement 2 implies statement 1” if yes then it is the correct reason.