7x – y + λ = 0

– 21 + 6 + λ = 0

λ = 15

7x – y + 15 = 0

x – y + k = 0

– 3 + 6 + k = 0 ⇒ k = – 3

x – y – 3 = 0

x – y – 3 = 0

7x – y – 5 = 0

__________                                                

– 6x + 2 = 0

 $\mathrm{x}=\frac{1}{3},\mathrm{y}=\frac{-5}{3}$

x – y + 1 = 0

7x – y + 15 = 0

___________            

– 6x – 14 = 0

$\mathrm{x}=\frac{-7}{3}$

$\mathrm{y}=\frac{-7}{3}+1=\frac{-4}{3}.$

Aliter:    Equation of angle bisector between

x – y + 1 = 0  and  7x – y – 5 = 0

$\frac{\mathrm{x}-\mathrm{y}+1}{\sqrt{2}}=\pm \frac{7\mathrm{x}-\mathrm{y}-5}{5\sqrt{2}}$

⇒ 5x – 5y + 5 = ± (7x – y – 5)

taking positive sign, x + 2y – 5 = 0

taking negative sign, 2x – y = 0

2x – y = 0 which passes through (–1, –2)

Another diagonal is

x + 2y + λ = 0 ⇒ –1 – 4 + λ = 0 Þ λ = 5

$\therefore$ x + 2y + 5 = 0

Now, solving x + 2y + 5 = 0 and 7x – y – 5 = 0

we get $(\frac{1}{3},\frac{-8}{3})$