QJEEM22Ma4100: JEE Main 2022 Mathematics - Class 12 JEEM22Ma4100 Solution: If the length of the perpendicular drawn from the point , a on…

$\because$ PR is perpendicular to given line, so

$2(2\lambda - 1 - a) + 3(3\lambda - 1) - 1( - \lambda - 1) = 0$

$\Rightarrow a = 7\lambda - 2$

Now,

$\because$ $PR = 2\sqrt 6$

$\Rightarrow {( - 5\lambda + 1)^2} + {(3\lambda - 1)^2} + {(\lambda + 1)^2} = 24$

$\Rightarrow 5{\lambda ^2} - 2\lambda - 3 = 0 \Rightarrow \lambda = 1$ or $- {3 \over 5}$

$\because$ a > 0 so $\lambda = 1$ and $a = 5$

Now $\sum\limits_{i = 1}^3 {{\alpha _i} = 2}$ (Sum of co-ordinate of R) $-$ (Sum of coordinates of P)

$= 2(7) - 11 = 3$

$a + \sum\limits_{i = 1}^3 {{\alpha _i} = 5 + 3 = 8}$