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