Question

Statement I: The point $A(3,1,6)$ is the mirror image of the point $P(1,3,4)$ in the plane $x-y+z=5$.

Statement II: The plane $x-y+z=5$ bisects the line segment joining $A(3,1,6)$ and $B(1,3,4)$ .

• A. Statement I is correct, Statement II is correct; Statement II is the correct explanation for Statement I
• B. Statement I is correct, Statement II is correct; Statement II is not the correct explanation for Statement I
• C. Statement I is correct, Statement II is incorrect
• D. Statement I is incorrect, Statement II is correct

Answer 1

The correct option is B

Statement I is correct, Statement II is correct; Statement II is not the correct explanation for Statement I

Explanation of correct option.

Finding the value of $a,b,$ and $c$ and finding the midpoint of the points given in statement II::

We can find the image of any point using the image formula. Hence, using the image formula for the point $\left(p,q,r\right)=(1,3,4)$ and the plane $lx+my+nz-s=0\Rightarrow x-y+z-5=0$

The image formula is $\frac{a-p}{l}=\frac{b-q}{m}=\frac{c-r}{n}=-2\left(\frac{lp+qm+rn+s}{\sqrt{l^2+m^2+n^2}}\right)$

$$\begin{gathered} \Rightarrow \frac{a-1}{1}=\frac{b-3}{-1}=\frac{c-4}{1}=-2\left(\frac{1-3+4-5}{3}\right) \\ \Rightarrow \frac{a-1}{1}=\frac{b-3}{-1}=\frac{c-4}{1}=2 \\ \therefore (a,b,c)=(3,1,6) \end{gathered}$$

Hence, Statement I is true.

Now,

The midpoint of the given points $\left(x_1,y_1,z_1\right)=A(3,1,6)$ and $\left(x_2,y_2,z_2\right)=B(1,3,4)$

The midpoint formula is $M=\left(\left(\frac{x_1+x_2}{2}\right),\left(\frac{y_1+y_2}{2}\right),\left(\frac{z_1+z_2}{2}\right)\right)$

$\left(\frac{3+1}{2},\frac{1+3}{2},\frac{4+4}{2}\right)=\left(2,2,5\right)$

Substituting the midpoint in $x-y+z=5$

$$\begin{gathered} x-y+z=5 \\ \Rightarrow 2-2+5=5 \\ \Rightarrow 5=5 \end{gathered}$$

Hence, Statement II is true.

Therefore, the correct answer is option (B).