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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2018 · Mathematics · Three Dimensional Geometry

What is the equation of the plane passing through the points...

What is the equation of the plane passing through the points (−2,6,−6),(−3,10,−9)(-2,6,-6),(-3,10,-9) and (−5,0,−6)(-5,0,-6) ?

Options

  1. A2x−y−2z=22 x-y-2 z=2
  2. B2x+y+3z=32 x+y+3 z=3
  3. Cx+y+z=6x+y+z=6
  4. Dx−y−z=3x-y-z=3

Correct answer

A. 2x−y−2z=22 x-y-2 z=2

Step-by-step solution

We know, the equation of plane passing through 3 points (x1,y1,z1),(x2,y2,z2)\left(\mathrm{x}_1, \mathrm{y}_1, \mathrm{z}_1\right),\left(\mathrm{x}_2, \mathrm{y}_2, \mathrm{z}_2\right) and \left(\mathrm{x}_3, \mathrm{y}_3, \mathrm{z}_3\right)isis\left|x−x1y−y1z−z1x2−x1y2−y1z2−z1x3−x1y3−y1z3−z1\begin{array}{ccc} \mathrm{x}-\mathrm{x}_1 & \mathrm{y}-\mathrm{y}_1 & \mathrm{z}-\mathrm{z}_1 \\ \mathrm{x}_2-\mathrm{x}_1 & \mathrm{y}_2-\mathrm{y}_1 & \mathrm{z}_2-\mathrm{z}_1 \\ \mathrm{x}_3-\mathrm{x}_1 & \mathrm{y}_3-\mathrm{y}_1 & \mathrm{z}_3-\mathrm{z}_1 \end{array}\right|=0So,theplanepassingthroughpointsSo, the plane passing through points(-2,6,-6),,(-3,10,-9)andand(-5,0,-6)isis∣x+2y−6z+6−14−3−3−60∣=0⇒(x+2)(−18)−(y−6)(−9)+(z+6)(6+12)=10⇒−18x−36+9y−54+18z+108=0⇒−18x+9y+18z+18=0⇒2x−y−2z−2=0⇒2x−y−2z=2\begin{aligned} & \left|\begin{array}{ccc} \mathrm{x}+2 & \mathrm{y}-6 & \mathrm{z}+6 \\ -1 & 4 & -3 \\ -3 & -6 & 0 \end{array}\right|=0 \\ & \Rightarrow(\mathrm{x}+2)(-18)-(\mathrm{y}-6)(-9)+(\mathrm{z}+6)(6+12)=10 \\ & \Rightarrow-18 \mathrm{x}-36+9 \mathrm{y}-54+18 \mathrm{z}+108=0 \\ & \Rightarrow-18 \mathrm{x}+9 \mathrm{y}+18 \mathrm{z}+18=0 \\ & \Rightarrow 2 \mathrm{x}-\mathrm{y}-2 \mathrm{z}-2=0 \\ & \Rightarrow 2 \mathrm{x}-\mathrm{y}-2 \mathrm{z}=2 \end{aligned}$
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