SATHEE: 3D Geometry 3 Question 61

3D Geometry 3 Question 61

####61. Consider the lines

$L_{1} : \frac{x - 1}{2} = \frac{y}{- 1} = \frac{z + 3}{1}, L_{2} : \frac{x - 4}{1} = \frac{y + 3}{1} = \frac{z + 3}{2}$ and the planes $P_{1} : 7 x + y + 2 z = 3, P_{2} : 3 x + 5 y - 6 z = 4 .$ Let $a x + b y + c z = d$ the equation of the plane passing through the point of intersection of lines $L_{1}$ and $L_{2}$ and perpendicular to planes $P_{1}$ and $P_{2}$ .

Match List I with List II and select the correct answer using the code given below the lists.

(2013 Adv.)

	List I		List II
P.	$a =$	1.	13
Q.	$b =$	2.	-3
R.	$c =$	3.	1
S.	$d =$	4.	-2

Codes

	$P$	$Q$	$R$	$S$		$P$	$Q$	$R$	$S$
(a)	3	2	4	1	(b) 1	3	4	2
(c)	3	2	1	4	(d) 2	4	1	3

Show Answer

Answer:

Correct Answer: 61. (a)

Solution:

$L_{1} : \frac{x - 1}{2} = \frac{y - 0}{- 1} = \frac{z - (- 3)}{1}$

$\begin{aligned} Normal of plane P : n & = | \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 7 & 1 & 2 \\ 3 & 5 & - 6 \end{array} | \\ = \hat{i} (- 16) - \hat{j} (- 42 - 6) + \hat{k} (32) \\ = - 16 \hat{i} + 48 \hat{j} + 32 \hat{k} \end{aligned}$

DR’s of normal $n = \hat{i} - 3 \hat{j} - 2 \hat{k}$

Point of intersection of $L_{1}$ and $L_{2}$ .

$\begin{aligned} \Rightarrow & 2 K_{1} + 1 & = K_{2} + 4 \\ and & - k_{1} & = k_{2} - 3 \\ \Rightarrow & k_{1} & = 2 and k_{2} = 1 \end{aligned}$

$∴$ Point of intersection $(5, - 2, - 1)$

Now equation of plane,

$\begin{aligned} 1 \cdot (x - 5) - 3 (y + 2) - 2 (z + 1) = 0 \\ \Rightarrow x - 3 y - 2 z - 13 = 0 \\ \Rightarrow x - 3 y - 2 z = 13 \\ ∴ a \to 1, b \to - 3, c \to - 2, d \to 13 \end{aligned}$

3D Geometry 3 Question 61

Codes

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

3D Geometry 3 Question 61

Codes

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu