First slide

Straight lines in 3D

Question

A plane P meets the coordinate axes at A, B and C respectively. The centroid of $Δ A B C$ is given to be (1, 1,2). Then the equation of the line through this centroid and perpendicular to the plane P is

Moderate

A

$\frac{x - 1}{1} = \frac{y - 1}{1} = \frac{z - 2}{2}$
B

$\frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2}$
C

$\frac{x - 1}{2} = \frac{y - 1}{2} = \frac{z - 2}{1}$
D

$\frac{x - 1}{2} = \frac{y - 1}{1} = \frac{z - 2}{1}$

Solution

$\begin{array}{l} Plane be \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \\ A = (a, 0, 0), B = (0, b, 0), C = (0, 0, c) \\ Centroid, G = (1, 1, 2) \end{array}$
$\begin{array}{l} \Rightarrow (\frac{a}{3}, \frac{b}{3}, \frac{c}{3}) = (1, 1, 2) \Rightarrow a = 3, b = 3, c = 6 \\ Plane is \frac{x}{3} + \frac{y}{3} + \frac{z}{6} = 1 \Rightarrow 2 x + 2 y + z = 6 \end{array}$
Dr’s of the line perpendicular to the plane are (2, 2, 1)
$equation of the line are \frac{x - 1}{2} = \frac{y - 1}{2} = \frac{z - 2}{1}$

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