Lines px+qy+r=0 ....(i) x‌sin‌α+y‌cos‌α=r...(ii) and ‌‌x‌cos‌α−y‌sin‌α=0...(iii) are intersecting at A ∴ lines are concurrent |
p
q
r
sin‌α
cos‌α
−r
cos‌α
−sin‌α
0
|=0 ⇒‌‌p‌sin‌α+q‌cos‌α+1=0 Angle between Eqs. (i) and (ii) is ‌
Ï€
3
∴tan‌
Ï€
3
=‌
‌
−p
q
+
sin‌α
cos‌α
1+‌
p‌sin‌α
q‌cos‌α
⇒√3=‌
−p‌cos‌α+q‌sin‌α
q‌cos‌α+p‌sin‌α
⇒√3(q‌cos‌α+p‌sin‌α)=−p‌cos‌α+q‌sin‌α −√3=−p‌cos‌α+q‌sin‌α[∵q‌cos‌α+p‌sin‌α=−1] ⇒p‌cos‌α−q‌sin‌α=√3 and p‌sin‌α+q‌cos‌α=−1