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Question
If the image of the point (2, 1) with respect to the line mirror be (5, 2), find the equation of the mirror.
Solution
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
\[y - \frac{3}{2} = - 3\left( x - \frac{7}{2} \right)\]
\[ \Rightarrow 2y - 3 = - 6x + 21\]
\[ \Rightarrow 6x + 2y - 24 = 0\]
\[ \Rightarrow 3x + y - 12 = 0\]
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