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Question

A ray of light is sent along the line which passes through the point $$(2,3)$$. The ray is reflected from the point $$P$$ on $$x-$$axis. If the reflected ray passes through the point $$(6,4)$$ then the coordinates of $$P$$ are


A
(267,0)
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B
(0,267)
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C
(267,0)
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D
None of these
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Solution

The correct option is A $$\displaystyle \left( \frac { 26 }{ 7 } ,0 \right) $$
Let $$P\equiv \left( \alpha ,0 \right) $$
Let the reflected ray makes an angle $$\theta$$ with +ve direction of x-axis, then the incident ray makes angle $$\left( \pi -\theta  \right) $$ with positive direction of x-axis.
The slope of the incident ray is 
$$\displaystyle =\frac { 0-3 }{ \alpha -2 } =\tan { \left( \pi -\theta  \right)  } \Rightarrow \tan { \theta  } =\frac { 3 }{ \alpha -2 } $$   ...(1)
The slope of the reflected ray is
$$\displaystyle =\frac { 4-0 }{ 6-\alpha  } =\tan { \theta  } \Rightarrow \tan { \theta  } =\frac { 4 }{ 6-\alpha  } $$   ...(2)
From (1) and (2), we get
$$\displaystyle \frac { 3 }{ \alpha -2 } =\frac { 4 }{ 6-\alpha  } \Rightarrow 18-3\alpha =4\alpha -8\Rightarrow 7\alpha =26\Rightarrow \alpha =\frac { 26 }{ 7 } $$
$$\therefore$$ The coordinate of $$P$$ are $$\displaystyle \left( \frac { 26 }{ 7 } ,0 \right) $$

389677_258340_ans_4e30871c4d174e09b6119cec26aad434.png

Mathematics

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