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JEE Main 2023 · Mathematics · Three Dimensional Geometry · medium

The distance of the point (1,1,1) from the plane passing through the points (1,0,0), (0,2,0) and (0,0,3) is

Single correct option

The distance of the point (1,1,1)(1,1,1) from the plane passing through the points (1,0,0)(1,0,0), (0,2,0)(0,2,0) and (0,0,3)(0,0,3) is
  1. A. 17\dfrac17
  2. B. 57\dfrac57
  3. C. 67\dfrac67
  4. D. 117\dfrac{11}{7}
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Answer: (B) 57\dfrac57

The plane with intercepts 1,2,31,2,3 is x1+y2+z3=1\frac x1+\frac y2+\frac z3=1, i.e. 6x+3y+2z−6=06x+3y+2z-6=0; each given point satisfies it. The normal has magnitude 36+9+4=7\sqrt{36+9+4}=7. Substituting (1,1,1)(1,1,1) into the left side gives 6+3+2−6=56+3+2-6=5. The distance is ∣5∣7=57\frac{|5|}{7}=\frac57. Dropping the constant term −6-6 gives the tempting wrong value 117\frac{11}{7}, and 67\frac67 is the distance of the origin from the plane.
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