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

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 $$\displaystyle \left( \frac { 26 }{ 7 } ,0 \right)$$Mathematics

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