Find the shortest distance between a line and a parabola

Question

Given the two curves,

$$ x - y - 3 = 0$$

$$ x + (y+2)^2 = 0$$

How do i find the shortest distance between the two?

My solution: find a point on the first line,

x1 = 6, y1 = 3

find the point on the 2nd line

x2 = -4, y2 = 0

but they are skewed, what is the formula then?

Answer 1

First, find the line that is tangential to the parabola and parallel to the given line, which has the form $y=x+m$. Plug it into $x +(y+2)^2=0$ to get

$$y^2+5y+4-m=0$$

Set its discriminant to zero since the line intersects with the parabola at only one point,

$$b^2-4ac = 9+4m=0$$

Solve for $m$,

$$m=-\frac94$$

Then, the shortest distant is just the distance between the two parallel lines, which is given by the distance formula,

$$d=\frac{|m-(-3)|}{\sqrt{1+k^2}}$$

where $k=1$ is the slope of the parallel lines. Thus,

$$d= \frac{3\sqrt{2}}{8}$$

Answer 2

Turning the problem into an optimization (calculus) problem is often a good trick. The word "shortest" in the question is your clue: you want to minimize a distance. Write down the distance you want to minimize, and then use calculus to find the minimum. Your distance should be a function of two variables.

Answer 3

The gradient of the parabola is $(1,2(y+2)$. The gradient must be perpendicular to the straight line, that is, is must be parallel to age straight line's gradient $(1, -2)$. Hence $y=-5/2$ and $x=-1/4$.