text: # If for a >0, the feet of perpendiculars from
  
  Question:
  
  If for a $>0$, the feet of perpendiculars from the points $\mathrm{A}(\mathrm{a},-2 \mathrm{a}, 3)$ and $\mathrm{B}(0,4,5)$ on the plane $l \mathrm{x}+\mathrm{my}+\mathrm{nz}=0$ are points $\mathrm{C}(0,-\mathrm{a},-1)$ and $\mathrm{D}$ respectively, then the length of line segment CD is equal to :
  
  1. $\sqrt{31}$
  
  2. $\sqrt{41}$
  
  3. $\sqrt{55}$
  
  4.  $\sqrt{66}$
  
  Correct Option: , 4
  
  Solution:
  
  C lies on plane $\Rightarrow-m a-n=0 \Rightarrow \frac{m}{n}=-\frac{1}{a} \ldots . .(1)$
  
  $\overrightarrow{\mathrm{CA}} \| \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+\mathrm{nk}$
  
  $\frac{\mathrm{a}-0}{l}=\frac{-\mathrm{a}}{\mathrm{m}}=\frac{4}{\mathrm{n}} \Rightarrow \frac{\mathrm{m}}{\mathrm{n}}=-\frac{\mathrm{a}}{4}$......(2)
  
  From (1) & (2)
  
  $-\frac{1}{a}=\frac{-a}{4} \Rightarrow a^{2}=4 \Rightarrow a=2 \quad($ since $a>0)$
  
  From (2) $\frac{\mathrm{m}}{\mathrm{n}}=\frac{-1}{2}$
  
  Let $m=-t \Rightarrow n=2 t$
  
  $\frac{2}{l}=\frac{-2}{-\mathrm{t}} \Rightarrow l=\mathrm{t}$
  
  So plane : $\mathrm{t}(\mathrm{x}-\mathrm{y}+2 \mathrm{z})=0$
  
  $\mathrm{BD}=\frac{6}{\sqrt{6}}=\sqrt{6} \quad \mathrm{C} \cong(0,-2,-1)$
  
  $\mathrm{CD}=\sqrt{\mathrm{BC}^{2}-\mathrm{BD}^{2}}$
  
  $=\sqrt{\left(0^{2}+6^{2}+6^{2}\right)-(\sqrt{6})^{2}}$
  
  $=\sqrt{66}$
  
  Saral
  Nov. 25, 2022, 1:52 a.m.
  explain Krna nahi bs answer chepna hai
explanation: The document contains a math question and its solution. The question asks for the length of the line segment CD given certain conditions about points A, B, C, and D, and a plane. The solution is provided, and the correct option is identified as 4, which corresponds to \(\sqrt{66}\).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # If for a >0, the feet of perpendiculars from
  
  Question:
  
  If for a $>0$, the feet of perpendiculars from the points $\mathrm{A}(\mathrm{a},-2 \mathrm{a}, 3)$ and $\mathrm{B}(0,4,5)$ on the plane $l \mathrm{x}+\mathrm{my}+\mathrm{nz}=0$ are points $\mathrm{C}(0,-\mathrm{a},-1)$ and $\mathrm{D}$ respectively, then the length of line segment CD is equal to :
  
  1. $\sqrt{31}$
  
  2. $\sqrt{41}$
  
  3. $\sqrt{55}$
  
  4.  $\sqrt{66}$
  
  Correct Option: , 4
  
  Solution:
  
  C lies on plane $\Rightarrow-m a-n=0 \Rightarrow \frac{m}{n}=-\frac{1}{a} \ldots . .(1)$
  
  $\overrightarrow{\mathrm{CA}} \| \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+\mathrm{nk}$
  
  $\frac{\mathrm{a}-0}{l}=\frac{-\mathrm{a}}{\mathrm{m}}=\frac{4}{\mathrm{n}} \Rightarrow \frac{\mathrm{m}}{\mathrm{n}}=-\frac{\mathrm{a}}{4}$......(2)
  
  From (1) & (2)
  
  $-\frac{1}{a}=\frac{-a}{4} \Rightarrow a^{2}=4 \Rightarrow a=2 \quad($ since $a>0)$
  
  From (2) $\frac{\mathrm{m}}{\mathrm{n}}=\frac{-1}{2}$
  
  Let $m=-t \Rightarrow n=2 t$
  
  $\frac{2}{l}=\frac{-2}{-\mathrm{t}} \Rightarrow l=\mathrm{t}$
  
  So plane : $\mathrm{t}(\mathrm{x}-\mathrm{y}+2 \mathrm{z})=0$
  
  $\mathrm{BD}=\frac{6}{\sqrt{6}}=\sqrt{6} \quad \mathrm{C} \cong(0,-2,-1)$
  
  $\mathrm{CD}=\sqrt{\mathrm{BC}^{2}-\mathrm{BD}^{2}}$
  
  $=\sqrt{\left(0^{2}+6^{2}+6^{2}\right)-(\sqrt{6})^{2}}$
  
  $=\sqrt{66}$
  
  Saral
  Nov. 25, 2022, 1:52 a.m.
  explain Krna nahi bs answer chepna hai
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its solution. The question asks for the length of the line segment CD given certain conditions about points A, B, C, and D, and a plane. The solution is provided, and the correct option is identified as 4, which corresponds to \(\sqrt{66}\).
  
  Label: question-and-answer