Let equation of line passing through A(−5,−4) is given by
x−(−5)
cos‌θ
=
y−(−4)
sin‌θ
=r ⇒
x+5
cos‌θ
=
y+4
sin‌θ
=r Therefore, every point on the line is of the form x=−5+r‌cos‌θ,y=−4+r‌sin‌θ Let AB=r1,AC=r2,AD=r3 ∵B=(−5+r1‌cos‌θ,−4+r1‌sin‌θ) and lies on x+3y+2=0 ⇒−5+r1‌cos‌θ+3(−4+r1‌sin‌θ)+2=0 ⇒r1=
15
cos‌θ+3‌sin‌θ
⇒
15
r1
=cos‌θ+3‌sin‌θ ...(i) ∵ C=(−5+r2‌cos‌θ,−4+r2‌sin‌θ) and lies on 2x+y+4=0 ⇒2(−5+r2‌cos‌θ)+(−4+r2‌sin‌θ)+4=0 ⇒
10
r2
=2‌cos‌θ+sin‌θ ...(ii) ∵ D=(−5+r3‌cos‌θ,−4+r3‌sin‌θ) and lies on x−y−5=0 ⇒−5+r3‌cos‌θ−(−4+r3‌sin‌θ)−5=0 ⇒
6
r3
=cos‌θ−sin‌θ ...(iii) Given that, (
15
AB
)2+(
10
AC
)2=(
6
AP
)2 ⇒(
15
r1
)2+(
10
r2
)2=(
6
r3
)2 ⇒(cos‌θ+3‌sin‌θ)2+(2‌cos‌θ+sin‌θ)2 =(cos‌θ−sin‌θ)2 [by Eqs. (i), (ii) and (iii)]