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NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2015 · Mathematics · Three Dimensional Geometry

A triangular plane ABC with centroid (1, 2, 3) cuts the coordinate...

A triangular plane ABC with centroid (1, 2, 3) cuts the coordinate axes at A, B, C respectively. What is the equation of plane ABC?

Options

  1. Ax+2y+3z=1x+2 y+3 z=1
  2. B3x+2y+z=33 x+2 y+z=3
  3. C2x+3y+6z=182 x+3 y+6 z=18
  4. D6x+3y+2z=186 x+3 y+2 z=18

Correct answer

D. 6x+3y+2z=186 x+3 y+2 z=18

Step-by-step solution

The plane passes through the point A(3,0,0),B(0,6,0)\mathrm{A}(3,0,0), \mathrm{B}(0,6,0) and C(0,0,9)C(0,0,9). So, it should satisfy the equation given in option for all the three points. From option (A) For point A (3,0,0)x+2y+3z=1(3,0,0)x+2 y+3 z=1⇒3+0+0≠1∴ option (A) is wrong. From option (B) For point A(3,0,0)3x+2y+z=3∴3(3)+0+0≠3∴ option (B) is wrong. From option (C) For point A(3,0,0)2x+3y+6z=18∴2(3)+0+0≠18∴ option (C) is wrong. From option (D)For point A(3,0,0)6x+3y+2z=18\begin{array}{ll}\Rightarrow & 3+0+0 \neq 1 \\ \therefore & \text{ option (A) is wrong. } \\ & \text{From option (B) } \\ & \text{For point } A(3,0,0) \\ & 3 \mathrm{x}+2 \mathrm{y}+\mathrm{z}=3 \\ \therefore & 3(3)+0+0 \neq 3 \\ \therefore & \text{ option (B) is wrong. } \\ & \text{From option (C) } \\ & \text{For point } \mathrm{A}(3,0,0) \\ & 2 \mathrm{x}+3 \mathrm{y}+6 \mathrm{z}=18 \\ \therefore \quad & 2(3)+0+0 \neq 18 \\ \therefore \quad & \text{ option }(\mathrm{C}) \text{ is wrong. } \\ & \text{From option }(\mathrm{D}) \\ & \text{For point } \mathrm{A}(3,0,0) \\ & 6 \mathrm{x}+3 \mathrm{y}+2 \mathrm{z}=18\end{array}⇒6(3)+0+0=18 For point B(0,6,0)6x+3y+2z=18∴0+3(6)+0=18 For point C(0,0,9)6x+3y+2z=180+0+2×9=18\begin{array}{ll} \Rightarrow& 6(3)+0+0=18 \\ \text{ For point } B(0,6,0) \\ & 6 x+3 y+2 z=18 \\ \therefore \quad & 0+3(6)+0=18 \\ \text{ For point } C(0,0,9) \\ & 6 x+3 y+2 z=18 \\ & 0+0+2 \times9=18 \end{array}∴\therefore \quad Option (D) is correct.
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