Question

Find the projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4).

Answer 1

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 \[\left( \frac{1}{5}, \frac{12}{5} \right)\].