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# The equation of the plane that meets the coordinate axes at the points $A,B,C$ respectively and the centroid of the $\Delta\:ABC$ is $(2,1,5)$ is ?

$(a)\:5x+10y+2z-30=0\:\:\qquad\:(b)\:2x+y+5z-30=0\:\:\qquad\:(c)\:5x+10y+2z+30=0\:\:\qquad\:(d)\:2x+y+5z+30=0$

Let the points on the coordinate axes, at which the plane meets be $A(a,0,0),\:\:B(0,b,0)\:\:and\:\:C(0,0,c)$
$\therefore$ The equation of the plane is $\large\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
Given that the centroid of the triangle $ABC=(2,1,5)$
$\Rightarrow\:\large\frac{a}{3}$$=2,\:\:\large\frac{b}{3}$$=1,\:\:\large\frac{c}{3}$$=5$
$\Rightarrow\:a=6,\:\:b=3\:\:and\:\:c=15$
$\Rightarrow\:$ The equation of the plane is $\large\frac{x}{6}+\frac{y}{3}+\frac{z}{15}=1$
$i.e.,\:\:5x+10y+2z=30$