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A triangular plane $\mathrm{ABC}$ with centroid $(1,2,3)$ cuts the coordinate axes at $A, B$, Crespectively. What is the equation of plane $A B C$ ?
Options:
Solution:
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Verified Answer
The correct answer is:
$6 x+3 y+2 z=18$
The plane passes through the point $\mathrm{A}(3,0,0), \mathrm{B}(0,6,0)$ and $\mathrm{C}(0,0,9)$. So, it should satisfy the equation given in option for all the three points. From option (a) For point $\mathrm{A}(3,0,0)$
$x+2 y+3 z=1$
$\Rightarrow 3+0+0 \neq 1$
option (a) is wrong. From option (b) For point $\mathrm{A}(3,0,0)$
$3 x+2 y+z=3$
$\therefore 3(3)+0+0 \neq 3$
$\therefore \quad$ option (b) is wrong. From option (c) For point $\mathrm{A}(3,0,0)$
$2 x+3 y+6 z=18$
$\therefore \quad 2(3)+0+0 \neq 18$
of option (c) is wrong. From option (d) For point $\mathrm{A}(3,0,0)$ $6 x+3 y+2 z=18$
$\Rightarrow 6(3)+0+0=18$
For point $\mathrm{B}(0,6,0)$
$6 x+3 y+2 z=18$
$\therefore \quad 0+3(6)+0=18$
For point $\mathrm{C}(0,0,9)$
$6 x+3 y+2 z=18$
$0+0+2 \times 9=18$
$\therefore \quad$ Option $(\mathrm{d})$ is correct.
$x+2 y+3 z=1$
$\Rightarrow 3+0+0 \neq 1$
option (a) is wrong. From option (b) For point $\mathrm{A}(3,0,0)$
$3 x+2 y+z=3$
$\therefore 3(3)+0+0 \neq 3$
$\therefore \quad$ option (b) is wrong. From option (c) For point $\mathrm{A}(3,0,0)$
$2 x+3 y+6 z=18$
$\therefore \quad 2(3)+0+0 \neq 18$
of option (c) is wrong. From option (d) For point $\mathrm{A}(3,0,0)$ $6 x+3 y+2 z=18$
$\Rightarrow 6(3)+0+0=18$
For point $\mathrm{B}(0,6,0)$
$6 x+3 y+2 z=18$
$\therefore \quad 0+3(6)+0=18$
For point $\mathrm{C}(0,0,9)$
$6 x+3 y+2 z=18$
$0+0+2 \times 9=18$
$\therefore \quad$ Option $(\mathrm{d})$ is correct.
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