Question

The distance between the two points P(x1,y1)P($$x_1$$,$$y_1) $$and R(x2,y2)R\left($$x_2$$,$$y_2$$\right) is ﻿d(P,R)=(x1−x2)2+(y1−y2)2d(P, R) = \sqrt{($$x_1$$ - $$x_2)^2$ + (y_1$ - y_2$)^2$$}
d(P,R)=(x1​−x2​)2+(y1​−y2​)2​﻿. Distance formula. Find the closest point on the line y=2xy = 2x to the point (1,7)(1, 7). (Hint: Every point on y=2xy = 2x has the form (x,2x)(x, 2x), and the closest point has the minimum distance.)

Accepted Answer

Identify the general form of any point on the line y = 2x, which is given as (x, 2x). Write the distance formula between the point (1, 7) and a general point on the line (x, 2x):
$$d = \sqrt{(x - 1)^2$ + (2x - 7)^2$}. $$Since the square root function is increasing, minimize the squared distance instead to simplify calculations:
$$d^2 = (x$$ - $$1)^2 + (2x$$ - $$7)^2. $$Expand and simplify the expression for $$d^2 to $$get a quadratic function in terms of $$x. $$Find the value of $$x$$ that minimizes $$d^2 by $$taking the derivative with respect to $$x$$, setting it equal to zero, and solving for $$x. $$Then, substitute this $$x$$ back into $$y = 2x to $$find the corresponding $$y$$ coordinate of the closest point.

Verified video answer for a similar problem:

Key Concepts

Distance Formula

Equation of a Line

Minimizing Distance Using Calculus or Algebra