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 : $\mathrm{t}(\mathrm{x}-\mathrm{y}+2 \mathrm{z})=0$

$\mathrm{BD}=\frac{6}{\sqrt{6}}=\sqrt{6} \quad \mathrm{C} \cong(0,-2,-1)$

$\mathrm{CD}=\sqrt{\mathrm{BC}^{2}-\mathrm{BD}^{2}}$

$=\sqrt{\left(0^{2}+6^{2}+6^{2}\right)-(\sqrt{6})^{2}}$

$=\sqrt{66}$

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Saral
Nov. 25, 2022, 1:52 a.m.
explain Krna nahi bs answer chepna hai
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