Question

The image of a point with respect to the line 2x-y+6=0 assuming the line to be a mirror is (6,6) find the point also find the equation of line joining this point and it's range

Answer 1

Answer:

The image of the given point is

$(\frac{-18}{5},\frac{54}{5})$

Step-by-step explanation:

Formula used:

The coordinates of the image of the point

$(x_1,y_1)$ with respect to the line ax+by+c=0 are given by

$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{-2(ax_1+by_1+c)}{a^2+b^2}$

Given line is 2x-y+6=0

Now using the above formula

$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{-2(ax_1+by_1+c)}{a^2+b^2}$

$\frac{x-6}{2}=\frac{y-6}{-1}=\frac{-2(2(6)-1(6)+6)}{2^2+(-1)^2}$

$\frac{x-6}{2}=\frac{y-6}{-1}=\frac{-2(12-6+6)}{4+1}$

$\frac{x-6}{2}=\frac{y-6}{-1}=\frac{-24}{5}$

Now

$\frac{x-6}{2}=\frac{-24}{5}$

$x-6=\frac{-48}{5}$

$x=6+\frac{-48}{5}$

$x=\frac{30-48}{5}$

$x=\frac{-18}{5}$

$\frac{y-6}{-1}=\frac{-24}{5}$

$y-6=\frac{24}{5}$

$y=6+\frac{24}{5}$

$y=\frac{30+24}{5}$

$y=\frac{54}{5}$

The image of the given point is

$(\frac{-18}{5},\frac{54}{5})$