Straight Lines and Pair of Straight Lines

Let theLmakes and angleθfrom positive direction ofX-axis.
∴The equation of L is
‌

x−1

cos‌θ

=‌

y−5

sin‌θ

=r‌ and ‌−r
LetA(r‌cos‌θ+1,rsin‌θ+5)andB(−r‌cos‌θ+1,−rsin‌θ+5)
be two points which lies in the given two lines5x−y−4=0and
3x+4y−4=0
because L bisects these lines.
‌5(r‌cos‌θ+1)−rsin‌θ−5−4=0
‌3(−r‌cos‌θ+1)+4(−rsin‌θ+5)−4=0
‌r(5‌cos‌θ−sin‌θ)=4....(i)
‌r(−3‌cos‌θ−4sin‌θ)=−19....(ii)
From Eqs. (i) and (ii)
‌−95‌cos‌θ+19sin‌θ=−12‌cos‌θ−16sin‌θ
‌⇒‌‌35sin‌θ=83‌cos‌θ
‌∴‌‌tan‌θ=‌

83

35

=m
∴ Equation of L is:
y−5=‌

83

35

(x−1)

⇒‌‌83x−35y+92=0