A line $L$ passing through origin is perpendicular to the lines
$L_{1} : \to r = (3 + t)^i + (- 1 + 2 t)^j + (4 + 2 t)^k$
$L_{2} : \to r = (3 + 2 s)^i + (3 + 2 s)^j + (2 + s)^k$
If the co-ordinates of the point in the first octant on $L_{2}$ at the distance of $\sqrt{17}$ from the point of intersection of $L$ and $L_{1}$ are $(a, b, c),$ then $18 (a + b + c)$ is equal to

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Solution

$L_{1} : \to r = (3 + t)^i + (- 1 + 2 t)^j + (4 + 2 t)^k$
$L_{1} : \frac{x - 3}{1} = \frac{y + 1}{2} = \frac{z - 4}{2}$
$\Rightarrow$ D.R..S. of $L_{1} = 1, 2, 2$
$L_{2} : \to r = (3 + 2 s)^i + (3 + 2 s)^j + (2 + s)^k$
$L_{2} : \frac{x - 3}{2} = \frac{y - 3}{2} = \frac{z - 2}{1}$
$\Rightarrow$ D.R.S. of $L_{2} = 2, 2, 1$
$∴$ D.R.S of required line L is parallel to D.R.S of $(L_{1} \times L_{2}) = (- 2, 3, - 2)$
$∴$ Equation of $L : \frac{x}{2} = \frac{y}{- 3} = \frac{z}{2}$
Solving $L & L_{1}$
$\Rightarrow (2 λ, - 3 λ, 2 λ) = (μ + 3, 2 μ - 1, 2 μ + 4)$
$\Rightarrow μ = - 1, λ = 1$
So, intersection point P(2, -3, 2)
Let, $Q (2 v + 3, 2 v + 3, v + 2)$ be required point on $L_{2}$
Now, $P Q = \sqrt{(2 v + 3 - 2)^{2} + (2 v + 3 + 3)^{2} + (v + 2 - 2)^{2}} = \sqrt{17}$
$\Rightarrow (2 v + 1)^{2} + (2 v + 6)^{2} + (v)^{2} = 17$
$\Rightarrow 9 v^{2} + 28 v + 20 = 0$
$\Rightarrow v = - 2$ (rejected), $\frac{10}{9}$ (accepted)
$∴ Q = (3 - \frac{20}{9}, 3 - \frac{20}{9}, 2 - \frac{10}{9})$
$= (\frac{7}{9}, \frac{7}{9}, \frac{8}{9})$
$∴ 18 (a + b + c)$
$= 18 (\frac{7}{9} + \frac{7}{9} + \frac{8}{9})$
$= 44$

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Similar questions

Q. A line

L

passing through origin is perpendicular to the lines

L_{1} : \to r = (3 + t)^i + (- 1 + 2 t)^j + (4 + 2 t)^k

L_{2} : \to r = (3 + 2 s)^i + (3 + 2 s)^j + (2 + s)^k

If the co-ordinates of the point in the first octant on

L_{2}

at the distance of

\sqrt{17}

from the point of intersection of

L

and

L_{1}

are

(a, b, c),

then

18 (a + b + c)

is equal to

A line l passing through the origin is perpendicular to the lines
$l_{1} : (3 + t)^i + (- 1 + 2 t)^j + (4 + 2 t)^k, - \infty < t < \infty l_{2} : (3 + 2 s)^i + (3 + 2 s)^j + (2 + s)^k, - \infty < s < \infty$
Then, the coordinate(s) of the point(s) on $l_{2}$ at a distance of $\sqrt{17}$ from the point of intersection of l and $l_{1}$ is (are)