Three Dimensional Geometry Question 29
Question: A variable plane is at a constant distance p from the origin and meets the axes in A, B and C. The locus of the centroid of the tetrahedron $ OABC $ is
Options:
A) $ {x^{-2}}+{y^{-2}}+{z^{-2}}=16{p^{-2}} $
B) $ {x^{-2}}+{y^{-2}}+{z^{-2}}=16{p^{-1}} $
C) $ {x^{-2}}+{y^{-2}}+{z^{-2}}=16 $
D) None of these
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Answer:
Correct Answer: A
Solution:
Plane is $ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 $ , where $ p=\frac{1}{\sqrt{\sum\limits_{{}}^{{}}{( \frac{1}{a^{2}} )}}} $ or $ \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{p^{2}} $ ?..(i) Now according to equation, $ x=\frac{a}{4},\ \ y=\frac{b}{4},\ \ z=\frac{c}{4} $ Put the values of x, y, z in (i), we get the locus of the centroid of the tetrahedron.
BETA
