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How to find the shortest distance from the point A to the line L?

A point A has the coordinate (-2, 0). A Line L has the equation y = 2x. A moving point P (x, y) move along the line y = 2x. Connect points A and P. In which location the point P move to so that the distance AP is the shortest?

a point move along the line L, find the shortest distance from point A(-2, 0) to the line L.

Solution

The shortest distance from the point A to the line L is that draw a line passing through the point A and perpendicular to the line L, the perpendicular line intersects the line L at the point P, then the distance AP is the shortest distance from point A to the line L.

A moving point move along the line L, find the shortest distance from the point A to the line L.

In the graph above, when the line segment AP is perpendicular to the line L, the distance AP is the shortest. Now, we need to find the coordinates x and y of the point P.

Two points determine a line. Draw the line L2 which pass through the two points A and P.

Since line L2 is perpendicular to line L, so, their slope are negative reciprocals of one another. Let K1 be the slope of L and K2 be the slope of L2. Then k2 = -1/k1. Since the slope of line y = 2x is 2, so the slope of L2 is -1/2. That is, K2 = -1/2.

Now we know the slope of L2 is -1/2 and the line L2 pass the point A (-2, 0). Using the point-slope line equation, y - yo = k2(x - x0), in which x0 = -2 and y0 = 0, k2 = -1/2. We get the line equation for L2. That is, y - 0 = -(1/2)[x - (-2)] = -(1/2)(x + 2), so, y = -x/2 - 1.

Since point P lies on both lines L and L2, so, point P (x, y) must satisfy both equations

y = 2x,
y = -x/2 - 1,
so, 2x = -x/2 - 1
2x + x/2 = -1
5x/2 = -1
5x = -2
x = -2/5
y = 2x = -4/5

so, when the coordinate of the point P is (-2/5, -4/5), AP is the shortest distance.