Find the point on the line y = 2x + 1 that is closest to (5, 2)

Solution:

The point on the line y = 2x + 1 which is closest to point (5, 2) is the point through which the line perpendicular to the line y = 2x + 1 passes through.

The first step is to determine the equation of the perpendicular line.

The slope of line y = 2x + 1 is 2.

The slope of the line perpendicular to line y = 2x + 1 will be -1/2

Since the perpendicular passes through the point (5, 2) also the equation of the perpendicular will be :

y - 2 = (-1/2)(x - 5)

y - 2 = -x/2 + 5/2

y = - x/2 + 9/2

Solving y = 2x + 1 and y = -x/2 + 9/2 simultaneously we will get the intersection point of the two lines and that will be the point closest to point (5, 2).

2x + 1 = -x/2 + 9/2

2x + x/2 = 9/2 - 1

5x/2 = 7/2

x = 7/5

y = 2(7/5) + 1 = 19/5

Therefore the point closest to (5, 2) and is on the line y = 2x + 1 is (7/5, 19/5)

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Find the point on the line y = 2x + 1 that is closest to (5, 2)

Summary:

The point on the line y = 2x+1 that is closest to (5, 2) is (7/5, 19/5).