A) 3
B) 1
C) 1/3
D) 9
Correct Answer: D
Solution :
[d] Let the equation of variable plane be |
\[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1.\] |
Which meets the axes at \[A(a,0,0),B(0,b,0)\] and\[C(0,0,c)\]. |
The centroid of \[\Delta ABC\] is \[\left( \frac{a}{3},\frac{b}{3},\frac{c}{3} \right)\]and it satisfies the relation\[\frac{1}{{{x}^{2}}}+\frac{1}{{{y}^{2}}}+\frac{1}{{{z}^{2}}}=k\]. Thus, |
\[\frac{9}{{{a}^{2}}}+\frac{9}{{{b}^{2}}}+\frac{9}{{{c}^{2}}}=k\] |
or \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}=\frac{k}{9}\] ??...(i) |
Also it is given that the distance of the plane \[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\] from (0, 0, 0) is 1 unit. Therefore, |
\[\frac{1}{\sqrt{\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}}}=1\] or \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}=1\] ?..(ii) |
From (i) and (ii), we get \[k/e=1,\] i.e., k=9 |
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