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Question
The reflection of the point (4, – 13) about the line 5x + y + 6 = 0 is ______.
Options
(– 1, – 14)
(3, 4)
(0, 0)
(1, 2)
Solution
The reflection of the point (4, – 13) about the line 5x + y + 6 = 0 is (– 1, – 14).
Explanation:
Let (h, k) be the point of reflection of the given point (4, – 13) about the line 5x + y + 6 = 0.
The mid-point of the line segment joining points (h, k) and (4, – 13) is given by
`(h + 4)/2, (k - 13)/2` (Why?)
This point lies on the given line, so we have
`5 (h + 4)/2 + (k - 13)/2 + 6` = 0
or 5 h + k + 19 = 0 .....(1)
Again the slope of the line joining points (h, k) and (4, –13) is given by `(k + 13)/(h - 4)`.
This line is perpendicular to the given line
Hence `(-5) (k + 3)/(h - 4)` = –1 (why?)
This gives 5k + 65 = h – 4
or h – 5k – 69 = 0 ....(2)
On solving (1) and (2)
We get h = –1 and k = –14.
Thus the point (–1, – 14) is the reflection of the given point.
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