Question

If the image of the point (2, 1) with respect to the line mirror be (5, 2), find the equation of the mirror.

Answer 1

Let the image of A (2, 1) be B (5, 2). Let M be the midpoint of AB.

\[\therefore \text { Coordinates of M } = \left( \frac{2 + 5}{2}, \frac{1 + 2}{2} \right)\]

                                       \[ = \left( \frac{7}{2}, \frac{3}{2} \right)\]

Let CD be the mirror.
The line AB is perpendicular to the mirror CD.

\[\therefore\] Slope of AB \[\times\] Slope of CD = −1

\[\Rightarrow \frac{2 - 1}{5 - 2} \times\text {  Slope of CD }= - 1\]

\[ \Rightarrow \text { Slope of CD} = - 3\]

Thus, the equation of the mirror CD is

\[y - \frac{3}{2} = - 3\left( x - \frac{7}{2} \right)\]

\[ \Rightarrow 2y - 3 = - 6x + 21\]

\[ \Rightarrow 6x + 2y - 24 = 0\]

\[ \Rightarrow 3x + y - 12 = 0\]