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Question
Write the coordinates of the image of the point (3, 8) in the line x + 3y − 7 = 0.
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Solution
Let the given point be A(3,8) and its image in the line x + 3y − 7 = 0 is B(h,k).
The midpoint of AB is \[\left( \frac{3 + h}{2}, \frac{8 + k}{2} \right)\] that lies on the line x + 3y − 7 = 0.
\[\therefore \frac{3 + h}{2} + 3 \times \frac{8 + k}{2} - 7 = 0\]
\[h + 3k + 13 = 0\] ... (1)
AB and the line x + 3y − 7 = 0 are perpendicular.
\[\therefore\text { Slope of AB } \times \text { Slope of the line } = - 1\]
\[ \Rightarrow \frac{k - 8}{h - 3} \times \frac{- 1}{3} = - 1\]
\[\Rightarrow 3h - k - 1 = 0\] ... (2)
Solving (1) and (2), we get:
(h, k) = (−1, −4)
Hence, the image of the point (3,8) in the line x + 3y − 7 = 0 is (−1,−4).
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