Question

Statement-1 : The point \[A\left( {3,{\text{ }}1,{\text{ }}6} \right)\] is the mirror image of the point \[B\left( {1,{\text{ }}3,{\text{ }}4} \right)\]in the plane \[x{\text{ }}-{\text{ }}y{\text{ }} + {\text{ }}z{\text{ }} = {\text{ }}5\]
Statement-2 : The plane x – y + z = 5 bisects the line segment joining A(3, 1, 6) and B(1, 3, 4).
(a) Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation of statement-1.
(b) Statement-1 is true, statement-2 is false.
(c) Statement-1 is false, statement-2 is true.
(d) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for statement-1.

Answer 1

Hint: First we will check if statement 1 is true or not by using the formula of image of a plane.
Then we will check if statement 2 is true or not by using the midpoint formula and then we will check if statement 1 and 2 are the correct explanation for each other or not.
The formula for image of a point \[\left( {x1,y1,z1} \right)\] with respect to the plane \[ax + by + cz + d = 0\] is given by:-
\[\dfrac{{x - {x_1}}}{a} = \dfrac{{y - {y_1}}}{b} = \dfrac{{z - {z_1}}}{c} = - 2\left( {\dfrac{{a{x_1} + b{y_1} + c{z_1} + d}}{{{a^2} + {b^2} + {c^2}}}} \right)\]
The midpoint \[\left( {x,y,z} \right)\] for two points \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\] is given by:-
\[x = \dfrac{{{x_1} + {x_2}}}{2}\]
\[y = \dfrac{{{y_1} + {y_2}}}{2}\]
\[z = \dfrac{{{z_1} + {z_2}}}{2}\]

Complete step-by-step answer:
Let us first check if statement 1 is true or not.
The given plane is
\[x{\text{ }}-{\text{ }}y{\text{ }} + {\text{ }}z{\text{ }} = {\text{ }}5\]
\[ \Rightarrow x{\text{ }}-{\text{ }}y{\text{ }} + {\text{ }}z{\text{ }} - {\text{ }}5 = 0\]
Hence comparing it with standard equation of plane \[ax + by + cz + d = 0\] we get:-
\[a = 1,b = - 1,c = 1,d = - 5\]
Now the given point whose image is to be calculated is \[B\left( {1,{\text{ }}3,{\text{ }}4} \right)\]
Hence,
\[{x_1} = 1,{y_1} = 3,{z_1} = 4\]
Now as we know that the formula for image of a point \[\left( {{x_1},{y_1},{z_1}} \right)\] with respect to the plane \[ax + by + cz + d = 0\] is given by:-
\[\dfrac{{x - {x_1}}}{a} = \dfrac{{y - {y_1}}}{b} = \dfrac{{z - {z_1}}}{c} = - 2\left( {\dfrac{{a{x_1} + b{y_1} + c{z_1} + d}}{{{a^2} + {b^2} + {c^2}}}} \right)\]
Hence substituting the values we get:-
\[\dfrac{{x - 1}}{1} = \dfrac{{y - 3}}{{ - 1}} = \dfrac{{z - 4}}{1} = - 2\left( {\dfrac{{\left( 1 \right)\left( 1 \right) + \left( { - 1} \right)\left( 3 \right) + \left( 1 \right)\left( 4 \right) - 5}}{{{{\left( 1 \right)}^2} + {{\left( { - 1} \right)}^2} + {{\left( 1 \right)}^2}}}} \right)\]
Soling it further we get:-
\[\dfrac{{x - 1}}{1} = \dfrac{{y - 3}}{{ - 1}} = \dfrac{{z - 4}}{1} = - 2\left( {\dfrac{{1 - 3 + 4 - 5}}{{1 + 1 + 1}}} \right)\]
\[\dfrac{{x - 1}}{1} = \dfrac{{y - 3}}{{ - 1}} = \dfrac{{z - 4}}{1} = - 2\left( {\dfrac{{ - 3}}{3}} \right)\]
\[\dfrac{{x - 1}}{1} = \dfrac{{y - 3}}{{ - 1}} = \dfrac{{z - 4}}{1} = 2\]
Now equating first and fourth term we get:-
\[\dfrac{{x - 1}}{1} = 2\]
Solving for x we get:-
\[x - 1 = 2\left( 1 \right)\]
\[ \Rightarrow x - 1 = 2\]
Simplifying for the value of x we get:-
\[ \Rightarrow x = 2 + 1\]
\[ \Rightarrow x = 3...................\left( 1 \right)\]
Similarly equating second and fourth term we get:-
\[\dfrac{{y - 3}}{{ - 1}} = 2\]
Solving for y we get:-
\[y - 3 = 2\left( { - 1} \right)\]
\[y - 3 = - 2\]
Simplifying for the value of y we get:-
\[y = - 2 + 3\]
\[y = 1.....................\left( 2 \right)\]
Now, equating third and fourth term we get:-
\[\dfrac{{z - 4}}{1} = 2\]
Solving for z we get:-
\[z - 4 = 2\left( 1 \right)\]
\[ \Rightarrow z - 4 = 2\]
Simplifying for the value of z we get:-
\[ \Rightarrow z = 4 + 2\]
\[ \Rightarrow z = 6......................\left( 3 \right)\]
From 1, 2 and 3 we get the image (x, y, z) as \[\left( {3,{\text{ }}1,{\text{ }}6} \right)\]
Hence,
 The point \[A\left( {3,{\text{ }}1,{\text{ }}6} \right)\] is the mirror image of the point \[B\left( {1,{\text{ }}3,{\text{ }}4} \right)\]in the plane \[x{\text{ }}-{\text{ }}y{\text{ }} + {\text{ }}z{\text{ }} = {\text{ }}5\]
Therefore, statement 1 is true.
Now let us check whether statement 2 is true or false.
Since we know that:
The midpoint \[\left( {x,y,z} \right)\] for two points \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\] is given by:-
\[x = \dfrac{{{x_1} + {x_2}}}{2}\]
\[y = \dfrac{{{y_1} + {y_2}}}{2}\]
\[z = \dfrac{{{z_1} + {z_2}}}{2}\]
The two given points are:
\[\left( {{x_1},{y_1},{z_1}} \right) = \left( {3,{\text{ }}1,{\text{ }}6} \right)\]
\[\left( {{x_2},{y_2},{z_2}} \right) = \left( {1,3,4} \right)\]
Hence applying midpoint formula on these points and substituting the values we get:-
\[ \Rightarrow x = \dfrac{{3 + 1}}{2}\]
\[ \Rightarrow x = \dfrac{4}{2}\]
\[ \Rightarrow x = 2\]
Now calculating for y we get:-
\[ \Rightarrow y = \dfrac{{1 + 3}}{2}\]
\[ \Rightarrow y = \dfrac{4}{2}\]
\[ \Rightarrow y = 2\]
Now calculating for z we get:-
\[ \Rightarrow z = \dfrac{{6 + 4}}{2}\]
\[ \Rightarrow z = \dfrac{{10}}{2}\]
\[ \Rightarrow z = 5\]
Hence the midpoint of A and B is \[\left( {x,y,z} \right) = \left( {2,2,5} \right)\]
Now in order to check if the plane bisects the line AB we need to check whether the midpoint of AB satisfies the equation of the plane or not.
The equation of the plane is:-
\[x{\text{ }}-{\text{ }}y{\text{ }} + {\text{ }}z{\text{ }} = {\text{ }}5\]
The midpoint of AB is \[\left( {x,y,z} \right) = \left( {2,2,5} \right)\]
Hence putting the midpoint in LHS of plane we get:-
\[LHS = \left( 2 \right) - \left( 2 \right) + 5\]
\[ \Rightarrow LHS = 0 + 5\]
\[ \Rightarrow LHS = 5\]
\[ \Rightarrow LHS = RHS\]
Now since \[LHS = RHS\]
Therefore the midpoint satisfies the plane.
Hence the plane x – y + z = 5 bisects the line segment joining A(3, 1, 6) and B(1, 3, 4).
Therefore statement 2 is true but that alone is not sufficient to determine whether A and B are mirror images of each other with respect to the plane.
Therefore, statement 2 is not the correct explanation of statement 1.

Hence, option 'A' is correct.

Note: In these types of questions we must check whether the given statements are true or not. If they are true we must see if there is any relation between the two statements. Students should take note that although statement 2 is true but the information given in this statement is not enough to explain statement 1.