Math, asked by gharshah8, 2 months ago

(4,
Find the direction cosines of two lines which are connected by the relations 1 – 5m +3n = 0
and 712 + 5m2 – 3n2 = 0.​

Answers

Answered by vinaysharmanobody
0

Answer:

Consider the given equation, l−5m+3n=0

l=5m−3n --------(1)

Substitute for l in 7l  

2

+5m  

2

−3n  

2

=0

7(5m−3n)  

2

+5m  

2

−3n  

2

=0

7(25m  

2

+9n  

2

−30mn)+5m  

2

−3n  

2

=0

175m  

2

+63n  

2

−210mn+5m  

2

−3n  

2

 

180m  

2

+60n  

2

−210mn

30(6m  

2

+2n  

2

−7mn)=0

6m  

2

+2n  

2

−7mn=0

(3m−2n)(2m−n)=0

Hence,

3m−2n=0   or   2m−n=0

Case (1) : 3m−2n=0 ⟹m=  

3

2n

​  

 

Substitute for m in (1)

l=5m−3n=5(  

3

2n

​  

)−3n=  

3

n

​  

 

Hence, the direction cosines (l,m,n) is (  

3

n

​  

,  

3

2n

​  

,n)

The direction ratio is proportional to (1,2,3) ----- {multiplying by 3}

We have the formula for Direction cosines, (  

a  

2

+b  

2

+c  

2

 

​  

 

a

​  

,  

a  

2

+b  

2

+c  

2

 

​  

 

b

​  

,  

a  

2

+b  

2

+c  

2

 

​  

 

c

​  

)

∴(  

1  

2

+2  

2

+3  

2

 

​  

 

1

​  

,  

1  

2

+2  

2

+3  

2

 

​  

 

2

​  

,  

1  

2

+2  

2

+3  

2

 

​  

 

3

​  

)=(±  

14

​  

 

1

​  

,±  

14

​  

 

2

​  

,±  

14

​  

 

3

​  

) are the direction cosines

Case (2) : 2m−n=0 ⟹m=  

2

n

​  

 

Substitute for m in (1)

l=5m−3n=5(  

2

n

​  

)−3n=−  

2

n

​  

 

Hence, the direction cosines (l,m,n) is (−  

2

n

​  

,  

2

n

​  

,n)

The direction ratio is proportional to (−1,1,2) ----- {multiplying by 2}

We have the formula for Direction cosines, (  

a  

2

+b  

2

+c  

2

 

​  

 

a

​  

,  

a  

2

+b  

2

+c  

2

 

​  

 

b

​  

,  

a  

2

+b  

2

+c  

2

 

​  

 

c

​  

)

∴(  

(−1)  

2

+1  

2

+2  

2

 

​  

 

−1

​  

,  

(−1)  

2

+1  

2

+2  

2

 

​  

 

1

​  

,  

(−1)  

2

+1  

2

+2  

2

 

​  

 

2

​  

)=(±  

6

​  

 

1

​  

,±  

6

​  

 

1

​  

,±  

6

​  

 

2

​  

) are the direction of coines

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