A ray of light is incident along a line which meets another line, `7x-y+1=0`, at the point (0,1). The ray is then reflected from this point along the line, `y+2x=1`. Then the equation of the line of incidence of the ray of light is:

Options

A. 41x + 38y - 38 = 0
B. 41 x - 38y + 38 = 0
C. 41x + 25y - 25 = 0
D. 41 x - 25y + 25 = 0

Answer

B

Explanation

The line along the incident ray passes through the intersection of the lines `7x - y + 1 = 0` and 2x + y - 1 = 0 . Let its equation be
`7 x - y + 1 + lambda (2x + y - 1) =0 " " … (i)`

Let P(-1,1) be a point on y + 2x = 1 . The image of P(1,-1) in the line mirror 7x - y + 1 = 0 is given by
`(x-1)/(7) = (y+1)/(-1) = -2 ((7xx1 - (-1) + 1))/( 7^(2) + (-1)^(2))`
`implies (x-1)/(7) = (y+1)/(-1) = - (18)/(50) implies x = - (38)/(25) , y = - (16)/(25)`
The line along the incident ray i.e. equation (i) passes through (-38/25 , -16/25) .
`therefore 7 (-(38)/(25)) + (16)/(25) + 1 + lambda (-(76)/(25) - (16)/(25) - 1) = 0 implies lambda = - (225)/(117)`
Substituting the value of `lambda` in (i) , we obtain
`7x - y + 1 - (225)/(117) (2x + y - 1) = 0 or , 41 x - 38 y + 38 = 0 `
as the line along the incident ray .