Reflected line

Geometry Level 5

A ray of light is sent along the line $2x-3y=5$ . After reflecting across the line $x+y=1$ it enters the opposite side by turning $15^{o}$ away from the line $x+y=1$ , The slope of line along which the reflected ray travels is

$m= \frac{ a \sqrt{b} - c } { d }$

where $a,b,c,d$ are co prime natural numbers, and $b$ is square free.

Find $a + b + c + d$ .

The answer is 37.

1 solution

Ayush Verma
May 5, 2015

Actually word 'refraction' should be used instead of 'reflection' to avoid the confusion.

Now the question is very simple,

$slope\quad of\quad 2x-3y=5\quad is\quad m=\tan { \theta =\cfrac { 2 }{ 3 } } \\ \\ as\quad ray\quad is\quad turning\quad away\quad by\quad { 15 }^{ o }\quad so\quad slope\quad of\quad refracted\quad ray\quad will\quad be\\ \\ { m }^{ ' }=\tan { (\theta +{ 15 }^{ o })= } \cfrac { \cfrac { 2 }{ 3 } +\left( 2-\sqrt { 3 } \right) }{ 1-\cfrac { 2 }{ 3 } \left( 2-\sqrt { 3 } \right) } \left( as\quad \tan { \theta =\cfrac { 2 }{ 3 } } \& \tan { { 15 }^{ o }=2-\sqrt { 3 } } \right) \\ \\ \quad =\cfrac { 8-3\sqrt { 3 } }{ 2\sqrt { 3 } -1 } =\cfrac {1 3\sqrt { 3 } -10 }{ 11} \\ \\ \Rightarrow a=13,b=3,c=10\& d=11\\ \\ \Rightarrow a+b+c+d=13+3+10+11=37$

I've updated the answer format. Can you update the solution accordingly?

Calvin Lin Staff - 6 years, 1 month ago

Reflected line

The answer is 37.

1 solution

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