Question

a ray of light is sent from the point (1,4).Upon reaching the x-axis,the ray is reflected from the point (3,0). this reflected ray is again reflected by the line x+y=5 and intersect y-axis at P. find the co-ordinate of p

Answer 1

Answer:

(0, -1/2)

Step-by-step explanation:

Hi,

Let the coordinates of P be ( 0, p).

Given that the ray of light is sent from point X(1, 4) and hits the x-xis at A(3,0)

Hence Slope of AX = (4-0)/(1-3) = -2

Thus, slope of AD will be -1*Slope of AX(laws of reflection)

Slope of AD = 2

Equation of AD will be (y - 0)/(x - 3) = 2

y = 2x - 6

Now AD intersects line x + y = 5 at D, hence the coordinates of D will be

( 11/3, 4/3)

Since slope of x + y = 5 is -1, the normal at D will have slope 1

Let slope of PD = m

Following laws of reflection, angle between the incident ray and normal = angle between normal and reflected ray,

Since incident ray has slope = 2

1 - m/(1 + m) = (2 - 1)/(1 + 2) = 1/3

⇒ 3 - 3m = m + 1

⇒ 4m  = 2

⇒ m = 1/2

Thus slope of PD is 1/2.

Hence Equation of PD will be

(y - 4/3) /(x - 11/3) = 1/2

⇒ 2y - 8/3 = x - 11/3

=> x - 2y = 1 is the equation of PD.

Point of intersection of line PD with y-axis is P(0, p)

Put x = 0 in x - 2y = 1 , we get

y = -1/2.

Hence , coordinates of P are ( 0 , -1/2).

Hope, it helped !