Find the point on the line y4x1y4x1 that is closest to the p

Find the point on the line y=4x+1y=4x+1 that is closest to the point (4,3)(4,3) .

Please show steps

Solution

y = 4x + 1 ==> Slope = dy/dx = 4

The closest point will between the line and the point (4, 3) will be on a perpendicular to the line.

A perpendicular line will have a slope that is the negative reciprocal of the slope of the original line, in this case, 1/4.

Basic equation of perpendicular line:

y = -1/4 x + b

Calculate b using the point (4, 3):

3 = -1 + b ==> b = 4

Equation of perpendicular line:

y = -1/4 x + 4

To find the point of intersection, set the two equations equal to each other and solve for x:

4x + 1 = -1/4 x + 4
4x + 1/4 x = 4 - 1 = 3

17/4 x = 3 ==> x = 12/17 = 0.70
then y = 4*(12/17) + 1 = 48/17 + 1 = 65/17 = 3.82

Point on line closest to (4, 3): (0.70, 3.82)