A light beam , emanating from the point A(3, 10) reflects from the straight line 2x + y - 6 - Maths - Straight Lines

Question

A light beam , emanating from the point A(3, 10) reflects from the
straight line 2x + y - 6 = 0 and then passes through the point B(7,
2) Find the equations of the incident and reflected beams

Rahul Raj · Accepted Answer

Dear student

Let the slope of the incident ray be m1 and reflected ray be m2

equation of these two lines are

y-10=m1(x-3)y-2=m2(x-7)point of interesection lies on 2x+y-6=0let this point be (a,6-2a)normal to 2x+y-6=0 has slope 1/2the incident and refracted beam make equal angles with this normalm1-1/21+m1×(1/2)=1/2-m21+m2×(1/2)simplifying we gwt3(m1+m2)+4m1m2-4=0y-10=m1(x-3) ; m1 = slope of line joining (3,10) and (a,6-2a)m1=6-2a-10a-3=-4-2aa-3y-2=m2(x-7)  ; m2 = slope of line joining (7,2) and (a,6-2a)m1=6-2a-2a-7=4-2aa-7putting value of m1 and m2 in 3(m1+m2)+4m1m2-4=0we get a=1m1=3 and m2=-1/3incident ray: m1=3, y-10=m1(x-3) or 3x-y+1=0reflected ray:  m2=-1/3, y-2=m2(x-7)  or x+3y-13=0

Regards

A light beam , emanating from the point A(3, 10) reflects from the straight line 2x + y - 6 = 0 and then passes through the point B(7, 2). Find the equations of the incident and reflected beams.