Image of point P with position vector $7\hat i - \hat j + 2\hat k$ in the line whose vector equation is $r = 9\hat i + 5\hat j + 5\hat k + \lambda (\hat i + 3\hat j + 5\hat k)$ has the position vector.
A. $( - 9,5,2)$
B. $(9,5, - 2)$
C. $(9, - 5, - 2)$
D. None

Hint: According to given in the question we have to determine the image of point P with position vector $7\hat i - \hat j + 2\hat k$ in the line whose vector equation is $r = 9\hat i + 5\hat j + 5\hat k + \lambda (\hat i + 3\hat j + 5\hat k)$ has the position vector. So, first of all we have to let all the three position vectors as some integer.
Now, we have to determine the position vector equation which can be determined with the help of the positions vectors we let.
Now, to determine the value of the position vectors we let, than we have to compare all the vectors as $\hat i,\hat j$ and $\hat k$ with the expression of the vector which is $r = 9\hat i + 5\hat j + 5\hat k + \lambda (\hat i + 3\hat j + 5\hat k)$ as mentioned in the question.
Now, as we know that the image of P which is $P'$ will be perpendicular to the equation of the given vector which is $r = 9\hat i + 5\hat j + 5\hat k + \lambda (\hat i + 3\hat j + 5\hat k)$
Now, we have to solve the obtained expression by substituting all the values in the expression vector which is $(\hat i + 3\hat j + 5\hat k)$ and make this equal to zero.

Complete step-by-step answer:
Step 1: First of all, we have to let all the three position vectors as some integer as mentioned in the solution hint. Hence,
$ \Rightarrow $Let and $\gamma $ are the position vectors.
Step 2: Now, we have to obtain the expression for the position vectors as we let in the solution step 1 with the help of the expression of position vector as mentioned in the question which is $7\hat i - \hat j + 2\hat k$. Hence,
$ = \dfrac{{\alpha + 7}}{2}\hat i + \dfrac{{\beta - 1}}{2}\hat j + \dfrac{{\gamma + 2}}{2}\hat k.....................(1)$
Step 3: Now, to determine the value of the position vectors we let, and then we have to compare all the vectors as $\hat i,\hat j$ and $\hat k$ with the expression of the vector which is $r = 9\hat i + 5\hat j + 5\hat k + \lambda (\hat i + 3\hat j + 5\hat k)$ as mentioned in the question which is as mentioned in the solution hint. Hence,
$ \Rightarrow \dfrac{{\alpha + 7}}{2}\hat i + \dfrac{{\beta - 1}}{2}\hat j + \dfrac{{\gamma + 2}}{2}\hat k = 9\hat i + 5\hat j + 5\hat k + \lambda (\hat i + 3\hat j + 5\hat k)$
Hence, on comparing $\hat i,\hat j$and $\hat k$in the both sides of the expressions,
\[
   \Rightarrow \dfrac{{\alpha + 7}}{2} = 9 + \lambda \\
   \Rightarrow \alpha = 18 + 2\lambda - 7 \\
   \Rightarrow \alpha = 11 + 2\lambda \\
 \]
Now, same as we have to determine the value of$\beta $,
\[
   \Rightarrow \dfrac{{\beta - 1}}{2} = 5 + 3\lambda \\
   \Rightarrow \beta = 10 + 6\lambda + 1 \\
   \Rightarrow \beta = 11 + 6\lambda \\
 \]
Now, same as we have to determine the value of$\gamma $,
\[
   \Rightarrow \dfrac{{\gamma + 2}}{2} = 5 + 5\lambda \\
   \Rightarrow \gamma = 10 + 10\lambda - 2 \\
   \Rightarrow \gamma = 8 + 10\lambda \\
 \]
Step 4: Now, as we know that the image of P which is $P'$ will be perpendicular to the equation of the given vector which is $r = 9\hat i + 5\hat j + 5\hat k + \lambda (\hat i + 3\hat j + 5\hat k)$as mentioned in the solution hint. Hence,
$ \Rightarrow (\alpha - 7) + 3(\beta + 1) + 5(\gamma - 2) = 0$
Now, on solving the vector equation as obtained just above,
$ \Rightarrow \alpha - 7 + 3\beta + 3 + 5\gamma - 10 = 0................(2)$
Step 5: Now, we have to substitute the values of $\alpha ,\beta $and $\gamma $ as we have already obtained in the solution step 3. Hence,
$
   \Rightarrow (11 + 2\lambda ) - 7 + 3(11 + 6\lambda ) + 3 + 5(8 + 10\lambda ) - 10 = 0 \\
   \Rightarrow 11 + 2\lambda - 7 + 33 + 18\lambda + 3 + 40 + 50\lambda - 10 = 0 \\
   \Rightarrow 70 + 70\lambda = 0 \\
   \Rightarrow 70(1 + \lambda ) = 0 \\
   \Rightarrow 1 + \lambda = 0 \\
   \Rightarrow \lambda = - 1 \\
 $
Step 6: Now, we have to substitute the value of $\lambda $ which we have already obtained in the solution step 5 to obtain the values of $\alpha ,\beta $ and $\gamma $. Hence,
$
   \Rightarrow \alpha = 11 + 2( - 1) \\
   \Rightarrow \alpha = 11 - 2 \\
   \Rightarrow \alpha = 9 \\
 $
Now, same as we have to determine the value of $\beta $
 $
   \Rightarrow \beta = 11 + 6( - 1) \\
   \Rightarrow \beta = 11 - 6 \\
   \Rightarrow \beta = 5 \\
 $
Now, same as we have to determine the value of $\gamma $
\[
   \Rightarrow \gamma = 8 + 10( - 1) \\
   \Rightarrow \gamma = 8 - 10 \\
   \Rightarrow \gamma = - 2 \\
 \]
Hence, image of point P is $(9,5, - 2)$

Hence, with the help of the position vectors we let we have determined the image of point P which is $(9,5, - 2)$. Therefore option (B) is correct.

Note:
It is necessary that we have to compare the obtained vector equation of the position vectors we let to determine the values of $\alpha ,\beta $ and $\gamma $. The image of P which is $P'$ will be perpendicular to the equation of the given vector which is $r = 9\hat i + 5\hat j + 5\hat k + \lambda (\hat i + 3\hat j + 5\hat k)$.