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Question
The coordinates of the foot of the perpendicular from the point (2, 3) on the line x + y – 11 = 0 are ______.
Options
(–6, 5)
(5, 6)
(–5, 6)
(6, 5)
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Solution
The coordinates of the foot of the perpendicular from the point (2, 3) on the line x + y – 11 = 0 are (5, 6).
Explanation:
Let (h, k) be the coordinates of the foot of the perpendicular from the point (2, 3) on the line x + y – 11 = 0.
Then, the slope of the perpendicular line is `(k - 3)/(h - 2)`
Again the slope of the given line x + y – 11 = 0 is – 1 (why?)
Using the condition of perpendicularity of lines, we have
`(k - 3)/(h - 2) (-1)` = – 1 (Why?)
or k – h = 1 ....(1)
Since (h, k) lies on the given line, we have,
h + k – 11 = 0 or h + k = 11 ....(2)
Solving (1) and (2)
We get h = 5 and k = 6.
Thus (5, 6) are the required coordinates of the foot of the perpendicular.
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