A plane meets the coordinate axes at A, B, an | vedclass

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(B) Let the equation of the plane be $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ $(i)$.Since the plane meets the coordinate axes at $A, B,$ and $C$,

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$\frac{x}{3} + \frac{y}{6} + \frac{z}{9} = 1$

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(B) Let the equation of the plane be $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ $(i)$.Since the plane meets the coordinate axes at $A, B,$ and $C$,the coordinates are $A(a, 0, 0)$,$B(0, b, 0)$,and $C(0, 0, c)$.The centroid of triangle $ABC$ is given by $\left( \frac{a+0+0}{3}, \frac{0+b+0}{3}, \frac{0+0+c}{3} \right) = \left( \frac{a}{3}, \frac{b}{3}, \frac{c}{3} \right)$.Given that the centroid is $(1, 2, 3)$,we have $\frac{a}{3} = 1$,$\frac{b}{3} = 2$,and $\frac{c}{3} = 3$.This gives $a = 3$,$b = 6$,and $c = 9$.Substituting these values into equation $(i)$,we get $\frac{x}{3} + \frac{y}{6} + \frac{z}{9} = 1$.

$A$ plane meets the coordinate axes at $A, B,$ and $C$ such that the centroid of triangle $ABC$ is $(1, 2, 3)$. Find the equation of the plane.

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