Let the image of the point P(1, 2, 3) in the line L:x - 6 3 = y - 1 2 = z - 2 3 be Q. Let R (, , ) be a point that divides internally the line segment PQ in the ratio 1 : 3. Then the value of 22 ( + + ) is equal to __________.

Let the image of the point P1 2 3 in the line Lx 6 3 y 1 2 z: (Mathematics)

The point dividing PQ in the ratio 1 : 3 will be mid-point of P & foot of perpendicular from P on the line.

$\therefore$ Let a point on line be $\lambda$

$\Rightarrow {{x - 6} \over 3} = {{y - 1} \over 2} = {{z - 2} \over 3} = \lambda$

$\Rightarrow P'(3\lambda + 6,\,2\lambda + 1,\,3\lambda + 2)$

as P' is foot of perpendicular

$(3\lambda + 5)3 + (2\lambda - 1)2 + (3\lambda - 1)3 = 0$

$\Rightarrow 22\lambda + 15 - 2 - 3 = 0$

$\Rightarrow \lambda = {{ - 5} \over {11}}$

$\therefore$ $P'\left( {{{51} \over {11}},{1 \over {11}},{7 \over {11}}} \right)$

Mid-point of $PP' \equiv \left( {{{{{51} \over {11}} + 1} \over 2},{{{1 \over {11}} + 2} \over 2},{{{7 \over {11}} + 3} \over 2}} \right)$

$\equiv \left( {{{62} \over {22}},{{23} \over {22}},{{40} \over {22}}} \right) \equiv (\alpha ,\beta ,\gamma )$

$\Rightarrow 22(\alpha ,\beta ,\gamma ) = 62 + 23 + 40 = 125$