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प्रश्न
A plane meets the coordinate axes at A, B and C, respectively, such that the centroid of triangle ABC is (1, −2, 3). Find the equation of the plane.
उत्तर
\[\text{ Let a,b and c be the intercepts of the given plane on the coordinate axes.} \]
\[\text{ Then the plane meets the coordinate axes at }\]
\[A \left( a, 0, 0 \right), B \left( 0, b, 0 \right) \text{ and C }\left( 0, 0, c \right)\]
\[\text{ Given that the centroid of the triangle is } \left( 1, - 2, 3 \right)\]
\[\Rightarrow\left( \frac{a + 0 + 0}{3}, \frac{0 + b + 0}{3}, \frac{0 + 0 + c}{3} \right)=\left( 1, - 2, 3 \right)\]
\[\Rightarrow\left( \frac{a}{3}, \frac{b}{3}, \frac{c}{3} \right)=\left( 1, - 2, 3 \right)\]
\[\Rightarrow\frac{a}{3}= 1,\frac{b}{3}= -2,\frac{c}{3}= 3\]
\[ \Rightarrow a = 3, b = - 6, c = 9 . . . \left( 1 \right)\]
\[\text{ Equation of the plane whose intercepts on the coordinate axes are a,b and c is } \]
\[\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\]
\[ \Rightarrow \frac{x}{3} + \frac{y}{- 6} + \frac{z}{9} = 1 [\text{ From } (1)]\]
\[ \Rightarrow 6x - 3y + 2x = 18\]
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