Let I₁ be the line 4 x+3 y=3 and I₂ be the line y=8 x . L₁ is the line formed by reflecting I₁ across the line y=x and L₂ is the line formed by reflecting I₂ across the x -axis. If is the acute angle between L₁ and L₂ such that = a b , where a and b are coprime then find ( a + b ) .

Step-by-step solution

I₁: 4 x+3 y=3f(x)=y= 3-4 x 3 ....(1) since f(x) and f⁻¹(x) are the mirror images of each other in the line y=x hence we find f⁻¹(x) . now y=f(x) f⁻¹(y)=x from (1) x= 3(1-y) 4 ; f⁻¹(y)= 3(1-y) 4 aligned & f⁻¹(x)= 3(1-x) 4 & 4 y=3-3 x L₁=3 x+4 y-3=0, m₁=-3 / 4 aligned |||ly L₂=y=-8 x with m₂=-8 if is the acute angle between the lines = | m ₂- m ₁ 1+ m ₁ ~m ₂ |= | -8+ 3 4 1+(-8) (- 3 4 ) |= | -29 28 | 29 28 aligned & a=29 and b=28 & a+b=29+28=57 aligned