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प्रश्न
A ray of light coming from the point (1, 2) is reflected at a point A on the x-axis and it passes through the point (5, 3). Find the co-ordinates of the point A
उत्तर
Let P(1, 2) and (5, 3) are the given points.
By the property of reflection,
∠XAB = ∠OAP = θ
(Angle of incidence = Angle of reflection)
Slope of the line OA(x – axis) m1 = 0
Slope of the line joining the points P(1, 2) and A(x, 0)
Slope of AP, m2 = `(2 - 0)/(1 - x) = 2/(1 - x)`
Slope of the line joining the points B(5, 3 ) and A(x, 0)
tan θ = `|("m"_1 + "m"_3)/(1 + "m"_1 "m"_3)|`
= `|(0 - 3/(5 - x))/(1 + 0(3/(5 - x)))|`
tan θ = `|(3/(5 - x))/1|`
= `3/(5 - x)` ......(1)
tan ∠XAB = `|("m"_1 + "m"_2)/(1 + "m"_1 "m"_2)|`
tan(180° – θ) = `|(0 - 2/(1 - x))/(1 - 0 xx 2/(1 - x))|`
– tan θ = `|(2/(1 - x))/1|`
= `2/(1 - x)`
tan θ = `- 2/(1 - x)` ......(2)
From equations (1) and (2)
`3/(5 - x) = - 2/(1 - x)`
3(1 – x) = – 2(5 – x)
3 – 3x = – 10 + 2x
2x + 3x = 10 + 3
5x = 13
⇒ x = `13/5`
∴ The required point A is `(13/5, 0)`
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