First slide

Planes in 3D

Question

If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B and C, then the locus of the centroid of $Δ A B C$ is

Moderate

A

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

$\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}} = 3$
C

$\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}} = \frac{1}{9}$
D

$\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}} = 9$

Solution

$A (a, 0, 0) B (0, b, 0) C (0, 0, c)$
$\Rightarrow G (x_{1}, y_{1}, z_{1}) = (\frac{a}{3}, \frac{b}{3}, \frac{c}{3})$
Equation of the plane is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ ; $\frac{x}{3 x_{1}} + \frac{y}{3 y_{1}} + \frac{z}{3 z_{1}} = 1$
$P = \frac{| d |}{\sqrt{a^{2} + b^{2} + c^{2}}}, 3 = \frac{1}{\sqrt{\frac{1}{9 x_{1}^{2}} + \frac{1}{9 y_{1}^{2}} + \frac{1}{9 z_{1}^{2}}}}$

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