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Question
A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.
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Solution
Let BC be the coordinate of that point A (a, 0) along the x-axis. AN is perpendicular to it. PA is an incident ray and AQ is a reflected ray.
⇒ Angle of incidence PAN = Angle of reflection NAQ
⇒ ∠PAB = ∠QAC
⇒ If the inclination of QA is 0 then the inclination of PA will be 180 – θ.
Slope of QA when Q(5, 3) and A(a, 0) then
`tan θ = (0 - 3)/("a" - 5) = (-3)/("a" - 5)`
Slope of PA when P(1, 2) and A(a, 0), then
tan (180° - θ) = `(0 - 2)/("a" - 1)`
tan (180° - θ) = − tan θ
∴ `(-2)/("a" - 1) = (-3)/("a" - 5) = 3/("a" - 5)`
or −2(a − 5) = 3(a − 1)
or −2a + 10 = 3a − 3
or 5a = 13
or a = `13/5`
∴ The coordinates of point A is `(13/5, 0)`.
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