Find the point on the line 2x + 3y + 5 = 0 which is closest to the point (5, 0).

Question

Find the point on the line 2x + 3y + 5 = 0 which is closest to the point (5, 0).

Write y as a linear function of the variable x.

y = -(2x+5)/3

Find the function f(x) that can be used to minimize this distance using the hint, if necessary. Remember, the root is not needed.

f(x) = ?

Find the first derivative of f(x) with respect to x.

Find the coordinates of the point on the line 2x + 3y + 5 = 0 that is closest to the point (5, 0).

Answer 1

Let Y1 = 2x + 3y + 5 = 0

Consider the point on the line which is closest to (5, 0) be (x,y)

Let the line formed by (5, 0) and (x, y) be Y2

Which indicates that the Y2 is perpendicular to the given line Y1 (Since, the minimum distance form a point on the line is the perpendiclar distance of from the point to the line)

slope of the line Y1 = -(2/3)

Therefore, slope of Y2 = -(1/Y1) = 3/2

Y2 = 3x - 2y - 15

(x, y) is the intersection point of Y1 and Y2

So, by solving Y1 and Y2, we wil be getting (x, y)

=> (x, y) = (35/13, -45/13)