First slide

Locus in 3D

Question

$If the points A (2 - x, 2, 2), B (2, 2 - y, 2), C (2, 2, 2 - z) and D (1, 1, 1) are coplanar, then$ $locus of P (x, y, z) is$

Moderate

A

$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$
B

$x + y + z = 1$
C

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

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

Solution

$Here \vec{A B} = \vec{O B} - \vec{O A =} x \hat{i} - y \hat{j}$
$\begin{array}{l} \vec{A C} = \vec{O C} - \vec{O A} = x \hat{i} - z \hat{k} \\ \vec{A D} = \vec{O D} - \vec{O A} = (x - 1) \hat{i} - \hat{j} - \hat{k} \\ As these vectors are coplanar, |\begin{array}{ccc} x & - y & 0 \\ x & 0 & - z \\ x - 1 & - 1 & - 1 \end{array}| = 0 \\ \Rightarrow x (- z) + y (- x + x z - z) = 0 \\ \Rightarrow x y + y z + z x = x y z \\ \Rightarrow \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1 \end{array}$

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