Find the point on the line which is closest to the point .

Question

Find the point on the line -4𝑥+6𝑦-4=0 which is closest to the point (4,5).

Please give the steps in a clear order.

Answer 1

One approach to take is to remember that the shortest distance from a given line to a specific point that doesn't lie on the line will be the perpendicular distance from the line.

By inspection of the coefficients for x and y, the given line has a slope of 2/3, so the perpendicular slope is -3/2 (the opposite reciprocal).

An equation for a the line with slope -3/2 passing thru (4,5) is y - 5 = -3/2(x - 4) or y = -3/2x + 11.

The point of intersection between that line and the given line can now be found by substitution:

-4x + 6(-3/2x + 11) - 4 = 0

-13x = -62

x = 62/13 , y = 50/13

An alternative, calculus-based approach, is to define the distance between any point on the given line and the given point in terms of a single variable, in this case x, and then to take the derivative of that distance function with respect to the variable and set it = 0:

y = 2/3x + 2/3

dist(x) = √((x - 4)² + (2/3x + 2/3 - 5)²) and to minimize this distance function it is sufficient to minimize the radicand:

d/dx [x² - 8x + 16 + 4/9x² - 52/9x + 169/9] = 26/9x - 124/9 = 0 ; x = 62/13