From the point (-1,-1), two rays are sent making angles of 45^ with the line x+y=0.

Question

From the point $(-1,-1)$, two rays are sent making angles of $45^{\circ}$ with the line $x+y=0$. These rays get reflected from the mirror $x+2 y=1$. If the equations of the reflected rays are $\mathrm{a} x+\mathrm{b} y=9$

Accepted Answer

Line $x+y=0$ has slope $-1$. If a ray has slope $m$ and makes an angle ${45}^{\circ }$ with this line, then by the angle formula between two lines we use:

$	an {45}^{\circ }=|\frac{m+1}{1-m}|$

Now $	an {45}^{\circ }=1$, so we get:

$	o \pm 1=\frac{m+1}{1-m}$

Solving this gives two possible directions. One solution gives a horizontal line ($m=0$), and the other corresponds to a vertical line (perpendicular to the $x$-axis).

$	o m=0$ \& one line is perpendicular to x axis.

Both rays start from $(-1,-1)$, so the two incident rays are:

${L}_{1}$: $x=-1$, ${L}_{2}$: $y=-1$

Next, each ray reflects from the mirror line $x+2y=1$. To get the reflected ray, we take the mirror image of the incident line in the mirror.

First, take the mirror image of $x+1=0$ in $x+2y=1$.

Now taking mirror image of $x+1=0$ in $x+2y=1$

${A}^{'}:\frac{x+1}{1}=\frac{y+1}{2}=\frac{-2(-1-2-1)}{{1}^{2}+{2}^{2}}$

From this, we get the image point:

$x=\frac{3}{5},y=\frac{11}{5}\, {A}^{'}(\frac{3}{5},\frac{11}{5})$

So the corresponding reflected line is written as:

line $=3x-4y+7=0$

Similarly, take the mirror image of $y+1=0$ in $x+2y=1$.

Now taking mirror image of $y+1=0$ in $x+2y=1$

${A}^{''}:\frac{x+1}{1}=\frac{y+1}{2}=\frac{-2(-1-2-1)}{{1}^{2}+{2}^{2}}$

This again gives:

$x=\frac{3}{5},y=\frac{11}{5}$

${A}^{''}(\frac{3}{5},\frac{11}{5})$

Now write the equation of the reflected line using point-slope form:

line :

$(y+1)=\frac{(\frac{11}{5}+1)}{(\frac{3}{5}-3)}(x-3)$

Simplifying gives one reflected ray as:

$4x+3y=9$

Compare $4x+3y=9$ with $ax+by=9$. Then:

comparing $4x+3y=9$ with $ax+by=9$ \&

$a=4$, $b=3$

The other reflected ray is written as:

by line

$-3x+4y=7$

Compare $-3x+4y=7$ with $cx+dy=7$. Then:

comparing with

$cx+dy=7$

$	herefore c=-3$, $d=4$

Now calculate $ad+bc$:

So, $ad+bc$

$=4	imes 4+3	imes (-3)$

$=16-9=7$

Solutions · 1