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 A (2, 1) be B (a, b). Let M be the midpoint of AB.

$\therefore \mathrm{Coordinates}\mathrm{of}M\mathrm{are}\equiv \left(\frac{2+a}{2},\frac{1+b}{2}\right)$



The point M lies on the line x + y − 5 = 0

$\therefore \frac{2+a}{2}+\frac{1+b}{2}-5=0$

$\Rightarrow a+b=7$ ... (1)

Now, the lines x + y − 5 = 0 and AB are perpendicular.

∴ Slope of AB $\times$ Slope of CD = −1

$$\begin{gathered} \Rightarrow \frac{b-1}{a-2}\times \left(-1\right)=-1 \\ \Rightarrow a-2=b-1 \end{gathered}$$
⇒ $a-b=1$ ... (2)

Adding eq (1) and eq (2):
$$\begin{gathered} 2a=8 \\ \Rightarrow a=4 \end{gathered}$$

Now, from equation (1):
$$\begin{gathered} 4+b=7 \\ \Rightarrow b=3 \end{gathered}$$

Hence, the image of the point (2, 1) with respect to the line mirror x + y − 5 = 0 is (4, 3).