9. The direction cosines of two lines aredetermined by the relation i-

1.	9. The direction cosines of two lines aredetermined by the relation i- 5m +3n 0 and7/2 +5m2-3n2-0, find them.
Answer» The given relations are l – 5m + 3n = 0⇒l = 5m – 3n……(1) and 7l^2+ 5m^2– 3n^2= 0 ……(2) Putting the value of l from (1) in (2), we get 7(5m – 3n)^2+ 5m^2– 3n^2= 0 or, 180m^2– 210mn + 60n^2= 0 or, (2m – n)(3m – 2n) = 0 ∴m/n = 1/2 or 2/3 whenm/n = 1/2 i.e. n = 2m ∴l = 5m – 3n = –m or1/m = –1 Hence, we have l/–1 = m/1 = n/2 = √ (l^2+ m^2+ n^2) /√ {(–1)^2+ l^2+ 22} = 1/√6 So, direction cosines of one line are –1/√6,1/√6,2/√6 Again whenm/n = 2/3 or n = 3m/2 ∴l = 5m – 3.3m/2 = m/2or 1/m = 1/2 Thus, m/n = 2/3and l/m = 1/2giving l/1 = m/2 = n/3 = 1/√1^2+ 2^2+ 3^2= 1/√14 The direction cosines of the other line are1/√14, 2/√14, 3/√14. Thanku so much

9. The direction cosines of two lines aredetermined by the relation i- 5m +3n 0 and7/2 +5m2-3n2-0, find them.

Answer»

The given relations are

l – 5m + 3n = 0⇒l = 5m – 3n……(1)

and 7l^2+ 5m^2– 3n^2= 0 ……(2)

Putting the value of l from (1) in (2), we get

7(5m – 3n)^2+ 5m^2– 3n^2= 0

or, 180m^2– 210mn + 60n^2= 0

or, (2m – n)(3m – 2n) = 0

∴m/n = 1/2 or 2/3

whenm/n = 1/2 i.e. n = 2m

∴l = 5m – 3n = –m or1/m = –1

Hence, we have l/–1 = m/1 = n/2 = √ (l^2+ m^2+ n^2) /√ {(–1)^2+ l^2+ 22} = 1/√6

So, direction cosines of one line are –1/√6,1/√6,2/√6

Again whenm/n = 2/3 or n = 3m/2

∴l = 5m – 3.3m/2 = m/2or 1/m = 1/2

Thus, m/n = 2/3and l/m = 1/2giving l/1 = m/2 = n/3 = 1/√1^2+ 2^2+ 3^2= 1/√14

The direction cosines of the other line are1/√14, 2/√14, 3/√14.

Thanku so much