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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2017 · Mathematics · Three Dimensional Geometry

(0,0,0),(a, 0,0),(0, b, 0) and (0,0, c) are four distinct points....

(0,0,0),(a,0,0),(0,b,0)(0,0,0),(\mathrm{a}, 0,0),(0, \mathrm{ b}, 0) and (0,0,c)(0,0, \mathrm{c}) are four distinct points. What are the coordinates of the point which is equidistant from the four points?

Options

  1. A(a+b+c3,a+b+c3,a+b+c3)\left(\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{3}, \frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{3}, \frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{3}\right)
  2. B(a,b,c)(a, b, c)
  3. C(a2, b2,c2)\left(\frac{\mathrm{a}}{2}, \frac{\mathrm{~b}}{2}, \frac{\mathrm{c}}{2}\right)
  4. D(a3, b3,c3)\left(\frac{\mathrm{a}}{3}, \frac{\mathrm{~b}}{3}, \frac{\mathrm{c}}{3}\right)

Correct answer

C. (a2, b2,c2)\left(\frac{\mathrm{a}}{2}, \frac{\mathrm{~b}}{2}, \frac{\mathrm{c}}{2}\right)

Step-by-step solution

A(0,0,0),B(a,0,0),C(0,b,0),D(0,0,c)\mathrm{A}(0,0,0), \mathrm{B}(\mathrm{a}, 0,0), \mathrm{C}(0, \mathrm{ b}, 0), \mathrm{D}(0,0, \mathrm{c}) Let the equidistant point be P(x,y,z)\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z}) i.e, AP=BP,AP=CP,AP=DP⇒(x−0)2+(y−0)2+(z−0)2=(x−a)2+y2+z2⇒x2+y2+z2=(x−a)2+y2+z2⇒x2=x2−2ax+a2⇒a2−2ax=0⇒a(a−2x)=0\begin{aligned} & \text{ i.e, } A P=B P, A P=C P, A P=D P \\ & \Rightarrow \sqrt{(x-0)^2+(y-0)^2+(z-0)^2} \\ & =\sqrt{(x-a)^2+y^2+z^2} \\ & \Rightarrow x^2+y^2+z^2=(x-a)^2+y^2+z^2 \\ & \Rightarrow x^2=x^2-2 a x+a^2 \\ & \Rightarrow a^2-2 a x=0 \\ & \Rightarrow a(a-2 x)=0 \end{aligned}$SinceSincea \neq 0, a=2 x \Rightarrow x=\frac{a}{2}Similarly,wewillgetSimilarly, we will get\mathrm{y}=\frac{\mathrm{b}}{2}, \mathrm{z}=\frac{\mathrm{c}}{2}$
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