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Question
Find the projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4).
Solution
Let A (−1, 2) be the given point whose projection is to be evaluated and C (−1, 2) and D (5, 4) be the other two points.
Also, let M (h, k) be the foot of the perpendicular drawn from A (−1, 2) to the line joining the points C (−1, 2) and D (5, 4).
Clearly, the slope of CD and MD are equal.
\[\therefore \frac{4 - k}{5 - h} = \frac{4 - 2}{5 + 1}\]
\[\Rightarrow h - 3k + 7 = 0\] ... (1)
The lines segments AM and CD are perpendicular.
\[\therefore\] \[\frac{k - 0}{h - 1} \times \frac{4 - 2}{5 + 1} = - 1\]
\[\Rightarrow 3h + k - 3 = 0\] ... (2)
Solving (1) and (2) by cross multiplication, we get:
\[\frac{h}{9 - 7} = \frac{k}{21 + 3} = \frac{1}{1 + 9}\]
\[ \Rightarrow h = \frac{1}{5}, k = \frac{12}{5}\]
Hence, the projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4) is \[\left( \frac{1}{5}, \frac{12}{5} \right)\].
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