Question

Find the image of the point $(2,1)$ with respect to the line mirror $x+y-5=0.$

Answer 1

Let the, image of the point $p(2,1)$ in the line mirror AB be $Q(\alpha ,\beta ).$ Then, PQ is perpendicularly bisected at R.

The coordinates of R are $(\frac{\alpha +2}{2},\frac{\beta +1}{2})$

And lie on the line $x+y-5=0$

$\left(\frac{\alpha +2}{2}\right)+\left(\frac{\beta +2}{2}\right)-5=0$

$\alpha +2+\beta +1-10=0$

$\alpha +\beta =7...\left(1\right)$

Since PQ is $\perp$ to AB

(Slope of AB) $\times$ (Slope of PQ) = - 1

$-1\times \left(\frac{\beta -1}{\alpha -2}\right)=-1$

$\beta -1=\alpha -2$

$\beta -\alpha =-1...\left(2\right)$

Solving (1) and (2), we get

$\alpha =4$ and $\beta =3$

$\therefore$ Image of $(1,2)$ in $x+y-5=0$ is $(4,3)$