Write an equation of the line that goes through a point (-2;-1) and cuts the segments in the third quadrant from the origin to x,y-axis intercepts such that these lengths of segments are minimum.

Answer 1

Let the xintercept be -2 - xinc and the yintercept be -1 -yinc (xinc and yinc will be positive).

As the slope of the line must be the same from (-2 - xinc, 0) to (-2, -1) and (0, -1 - yinc),

-1/xinc = -yinc/2; note how defining the xintercepts as we did made this an easy equation.

xinc yinc = 2

We want to minimize 2 + xinc + 1 + yinc = 3 + xinc + yinc, or equivalently xinc + yinc

From the AM-GM inequality, as xinc yinc = 2, this is minimized when xinc = √2 and yinc = √2

The slope of the line is -yinc/2 = -√2/2

As the y-intercept is -1 - yinc = -1 - √2, the equation of the line is

y = -√2/2 x -1 - √2

We can check that this goes through the x-intercept and (-2, -1)

0 = -√2/2 x -1 - √2

√2/2 x = -1 - √2

x = √2(-1 - √2) = -2 - √2; this equals -2 - xinc

For (-2, -1), mx + b = -√2/2 x -1 - √2 = -√2/2(-2) -1 - √2 = √2 -1 - √2 = -1

y = -√2/2 x -1 - √2