A ray of light is sent along the line x2y3=0. Upon reaching the line3x2y5=0, the ray is reflected from it. If the equation of the line containing reflected ray is ax2y=b, then find the value of(a+b)?

1 Answer
Sep 13, 2017

a+b=60

Explanation:

We are given 3 equations:

x2y3=0 [1]
3x2y5=0 [2]
ax2y=b [3]

Find the slope of the line corresponding to equation [1]

2y=x+3

y=12x32

m1=12

The angle that it forms with the x axis is θ1=tan1(12)

Find the slope of the line corresponding to equation [2]:

2y=3x+5

y=32x52

m2=32

The angle that it forms with the x axis is θ2=tan1(32)

The angle from line [1] to line [2] is the angle of incidence:

θi=tan1(32)tan1(12) [4]

Find the slope of the line corresponding to equation [3]:

2y=ax+b

y=a2xb2

m3=a2

The angle that it forms with the x axis is θ3=tan1(a2)

The angle from line [2] to line [3] is the angle of reflection:

θr=tan1(a2)tan1(32) [5]

Because the angle of incidence equals the angle of reflection we can set the right side of equation [4] equal to the right side of equation 5:

tan1(a2)tan1(32)=tan1(32)tan1(12)

tan1(a2)=2tan1(32)tan1(12)

a=2tan(2tan1(32)tan1(12))

a=29

Substitute into equation [3]:

29x2y=b [3.1]

Find the point of intersection of lines [1] and [2]:

x2y3=0 [1]
3x2y5=0 [2]

Subtract [1] from [2]:

2x2=0

x=1

Substitute 1 for x into equation [1]:

12y3=0

#-2y -2 = 0

y=1

The point is (1,1)

Equation [3.1] must contain the same point:

29(1)2(1)=b

b=31

a+b=60