First slide

Straight lines in 3D

Question

$The line passing through the points (5, 1, a) and (3, b, 1) crosses the y z -plane at the point (0, \frac{17}{2}, \frac{- 13}{2}) then$

Moderate

A

$a = 2, b = 8$
B

$a = 4, b = 6$
C

$a = 6, b = 4$
D

$a = 8, b = 2$

Solution

Equation of the line passing through $(5, 1, a) a n d (3, b, 1) i s \frac{x - 5}{2} = \frac{y - 1}{1 - b} = \frac{z - a}{a - 1}$
Let $\frac{x - 5}{2} = \frac{y - 1}{1 - b} = \frac{z - a}{a - 1} = λ (s a y)$
$\Rightarrow x = 2 λ + 5, y = λ (1 - b) + 1, z = λ (a - 1) + a T h e l i n e m e e t s y z - p l a n e a t (0, \frac{17}{2}, - \frac{13}{2}) \Rightarrow 2 λ + 5 = 0, λ (1 - b) + 1 = \frac{17}{2} a n d λ (a - 1) + a = - \frac{13}{2} \Rightarrow λ = - \frac{5}{2} a n d b = 4, a = 6$

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