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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2023 · Mathematics · Three Dimensional Geometry

If L is the line with DIRECTIONS ratios and passing through...

If LL is the line with DIRECTIONS ratios <3,−2,6>< 3,-2,6 > and passing through (1,−1,1)(1,-1,1), then what are the coordinates of the points on L whose distance from (1,−1,1)(1,-1,1) is 2 units?

Options

  1. A(−117,137,197)\left(-\frac{11}{7}, \frac{13}{7}, \frac{19}{7}\right) and (17,37,57)\left(\frac{1}{7}, \frac{3}{7}, \frac{5}{7}\right)
  2. B(197,−117,137)\left(\frac{19}{7},-\frac{11}{7}, \frac{13}{7}\right) and (−17,37,−57)\left(-\frac{1}{7}, \frac{3}{7},-\frac{5}{7}\right)
  3. C(137,117,197)\left(\frac{13}{7}, \frac{11}{7}, \frac{19}{7}\right) and (−17,−37,57)\left(-\frac{1}{7},-\frac{3}{7}, \frac{5}{7}\right)
  4. D(137,−117,197)\left(\frac{13}{7},-\frac{11}{7}, \frac{19}{7}\right) and (17,−37,−57)\left(\frac{1}{7},-\frac{3}{7},-\frac{5}{7}\right)

Correct answer

D. (137,−117,197)\left(\frac{13}{7},-\frac{11}{7}, \frac{19}{7}\right) and (17,−37,−57)\left(\frac{1}{7},-\frac{3}{7},-\frac{5}{7}\right)

Step-by-step solution

Since equation of line is x−13=y+1−2=z−16=λ (let) ⇒(x,y,z)=(3λ+1,−2λ−1,6λ+1)\begin{aligned} & \frac{x-1}{3}=\frac{y+1}{-2}=\frac{z-1}{6}=\lambda \text{ (let) } \\ & \Rightarrow(x, y, z)=(3 \lambda+1,-2 \lambda-1,6 \lambda+1) \end{aligned} Now, (3λ+1−1)2+(−2λ−1+1)2+(6λ+1−1)2=2=9λ2+4λ2+36λ2=2⇒49λ2=2⇒±7λ=2⇒λ=±27\begin{aligned} & \sqrt{(3 \lambda+1-1)^2+(-2 \lambda-1+1)^2+(6 \lambda+1-1)^2}=2 \\ & =\sqrt{9 \lambda^2+4 \lambda^2+36 \lambda^2}=2 \\ & \Rightarrow \sqrt{49 \lambda^2}=2 \Rightarrow \pm 7 \lambda=2 \Rightarrow \lambda= \pm \frac{2}{7} \end{aligned} So, (x,y,z)=(x, y, z)=\left(\frac{13}{7}, \frac{11}{7}, \frac{19}{7}\right)oror\left(17\frac{1}{7}, −37\frac{-3}{7}, \frac{-5}{7}\right)
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