Mathematics

# If two line $L_1$ and $L_2$ in space,are defined by$\begin{array}{l}{L_1} = \left\{ {x = \sqrt \lambda y + \left( {\sqrt \lambda - 1} \right),z = \left( {\sqrt \lambda - 1} \right)y + \sqrt \lambda } \right\}and\\{L_2} = \left\{ {x = \sqrt \mu y + \left( {1 - \sqrt \mu } \right),z = \left( {1 - \sqrt \mu } \right)y + \sqrt \mu } \right\}\end{array}$, then $L_1$ is perpendicular to $L_2$ for all non-negative reals $\lambda$ and $\mu$, such that:

##### ANSWER

$\sqrt \lambda + \sqrt \mu = 0$

##### SOLUTION
$\begin{array}{l} { L_{ 1 } }=\{ x=\sqrt { \lambda } y+(\sqrt { \lambda } -1),z=(\sqrt { \lambda } -1)y+\sqrt { \lambda } \} \\ \dfrac { { x-(\sqrt { \lambda } -1) } }{ { \sqrt { \lambda } } } =y \\ \dfrac { { z-\sqrt { \lambda } } }{ { (\sqrt { \lambda } -1) } } =y \\ \dfrac { { x-(\sqrt { \lambda } -1) } }{ { \sqrt { \lambda } } } =y=\dfrac { { z-\sqrt { \lambda } } }{ { (\sqrt { \lambda } -1) } } \\ { L_{ 1 } }=\left( { \sqrt { \lambda } -1,0,\sqrt { \lambda } } \right) +{ K_{ 1 } }\left( { \sqrt { \lambda } ,1,\sqrt { \lambda } -1 } \right) \\ Now, \\ { L_{ 2 } }=\{ x=\sqrt { \mu } y+(1-\sqrt { \mu } ),z=(1-\sqrt { \mu } )y+\sqrt { \mu } \} \\ \dfrac { { x-(1-\sqrt { \mu } ) } }{ { \sqrt { \mu } } } =y \\ \dfrac { { z-\sqrt { \mu } } }{ { 1-\sqrt { \mu } } } =y \\ { L_{ 2 } }=\left( { 1-\sqrt { \mu } ,0,\sqrt { \mu } } \right) +{ K_{ 2 } }\left( { \sqrt { \mu } ,1,1-\sqrt { \mu } } \right) \\ { L_{ 1 } }\bot { L_{ 2 } } \\ \left( { \sqrt { \lambda } ,1,\sqrt { \lambda } -1 } \right) .\left( { \sqrt { \mu } ,1,1-\sqrt { \mu } } \right) =0 \\ \sqrt { \lambda \mu } +1+\left( { \sqrt { \lambda } -1 } \right) \left( { 1-\sqrt { \mu } } \right) =0 \\ \sqrt { \lambda \mu } +1+\sqrt { \lambda } -1-\sqrt { \lambda \mu } +\sqrt { \mu } =0 \\ \sqrt { \lambda } +\sqrt { \mu } =0 \\ \ \end{array}$

You're just one step away

Create your Digital Resume For FREE on your name's sub domain "yourname.wcard.io". Register Here!

Single Correct Medium Published on 09th 09, 2020
Questions 120418
Subjects 10
Chapters 88
Enrolled Students 86

#### Realted Questions

Q1 Subjective Medium
In figure, lines $l_1$ and $l_2$ intersect at $O$, forming angles as shown in the figure. If $x=45^o$, find the values of $y, z$ and $u$.

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q2 Single Correct Medium
OA and OB are opposite rays.
If $x=75^o$, what is the value of y?
• A. $75^o$
• B. $45^o$
• C. $15^o$
• D. $105^o$

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q3 Single Correct Medium
The supplementry angle of an angle is one third of itself. Then the angle of its supplement are
• A. $60^0,80^0$
• B. $120^0,360^0$
• C. $60^0,120^0$
• D. $135^0,45^0$

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q4 Single Correct Medium
Two supplementary angles are in ratio $4:5$. Find the measure of greater angle.
• A. $\displaystyle { 70 }^{ o }$
• B. $\displaystyle { 80 }^{ o }$
• C. $\displaystyle { 110 }^{ o }$
• D. $\displaystyle { 100 }^{ o }$

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q5 Subjective Medium
From the adjoining figure find $x$,  i $AOB$ is a straight line. Hence complete the following
Which is an acute angle?

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020