If the points $(- 1, 2, - 3), (4, a, 1)$ and $(b, 8, 5)$ are collinear, then a and b are respectively equal to

$5$ and $5$
$9$ and $5$
$5$ and $9$
$- 5$ and $9$

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Solution

The correct option is C
$5$ and $9$

Step1: Find $\vec{A B}$ and $\vec{A C}$
Let the given points be $A (- 1, 2, - 3), B (4, a, 1)$ and $C (b, 8, 5)$
$\vec{A B} = <(4 - (- 1)), (a - 2), (1 - (- 3))> = <5, a - 2, 4>$
$\vec{A B} = <(4 - (- 1)), (a - 2), (1 - (- 3))> = <5, a - 2, 4>$
Since, the points are collinear.
$x 2 - x 1 = x 3 - x 2, y 2 - y 1 = y 3 - y 2$
=> $b - 4 = 4 - (- 1)$ and $a - 2 = 8 - a$
=> $b = 9$ and $2 a = 10$
=> $a = 5$ and $b = 9$
Hence, option $(3)$ is correct.

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