If the image of the point $\mathrm{P}(a, 2, a)$ in the line $\frac{x}{2}=\frac{y+a}{1}=\frac{z}{1}$ is Q and the image of Q in the line $\frac{x-2 b}{2}=\frac{y-a}{1}=\frac{z+2 b}{-5}$ is P, then $a+b$ is equal to $\_\_\_\_$.

Solution

Now,

$\dfrac{2\lambda - a + a}{2} = -5\mu - 2b$
and $\dfrac{a + 4\lambda - a}{2} = 2\mu + 2b$
and $\dfrac{2\lambda - 2a - 2 + 2}{2} = \mu + a$
$\Rightarrow \lambda - \mu = b$
$\lambda - \mu = 2a$
$\lambda + 5\mu = -2b$
$\Rightarrow b = 2a$ and $\lambda = a$, $\mu = -a$

$(a\hat{i} - 2\hat{j} + 0\hat{k}) \cdot (2\hat{i} + \hat{j} + \hat{k}) = 0$
$2a - 2 = 0$
$a = 1$
$b = 2a = 2 \times 1 = 2$
$a + b = 1 + 2 = 3$

Hence, the correct answer is 3.