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Question

A ray of light coming from the point $$(1, 2)$$ is reflected at a point $$A$$ on the x-axis and then passes through the point $$(5, 3)$$. The coordinates of the point $$A$$ are


A
(135,0)
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B
(513,0)
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C
(7,0)
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D
None of these
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Solution

The correct option is B $$\left ( \displaystyle \frac{13}{5},0 \right )$$
Let the coordinates of point A be $$(a,0)$$
Let OL be the normal .
$$\Rightarrow \angle BAL=\angle CAL =\phi$$
Let $$\angle CAX=\theta$$
$$\Rightarrow \angle OAB=\theta$$
So, $$\angle BAX={90}^{0}+\phi$$
Slope of line Ac $$=\displaystyle \tan  ({ 90 }^{ 0 }-\phi )=\frac { 3 }{ 5-a } $$
$$\Rightarrow \cot  \phi =\dfrac { 3 }{ 5-a } $$                ......(i)
Slope of line AB $$=\displaystyle \tan  ({ 90 }^{ 0 }+\phi )=\frac { 2 }{ 1-a } $$
$$\Rightarrow \cot  \phi =\dfrac { 2 }{ a-1 } $$                ......(ii)
From (i) and (ii),

$$\Rightarrow \dfrac { 3 }{ 5-a } =\dfrac { 2 }{ a-1 } $$

$$\Rightarrow a=\dfrac { 13 }{ 5 } $$

Hence, the image is 
$$(\dfrac { 13 }{ 5 } ,0)$$

131172_124034_ans_21e93e07811f4fb79dc67f92e0d751f9.png

Physics

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