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Given the two curves,

$$ x - y - 3 = 0$$

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

How do i find the shortest distance between the two?

kk

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?

Rodrigo de Azevedo
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ERJAN
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  • Are you allowed to use the diagram or plotting device? – NoChance Sep 13 '19 at 21:26
  • @NoChance, yes i m – ERJAN Sep 13 '19 at 21:27
  • but the general formula and uderstanding would be better... – ERJAN Sep 13 '19 at 21:27
  • Usually problems like this are solved using Calculus, not linear algebra. Do you know any Calculus ? – WW1 Sep 13 '19 at 21:27
  • @WW1, i do know calc, i just did not refresh it for long time... – ERJAN Sep 13 '19 at 21:28
  • If so, it is may be possible to spot it.I guess the shortest distance has to be perpendicular to the line. We also need to take know the equation of the tangent to the parabola. I guess it is not trivial. Also, this may be related:https://math.stackexchange.com/questions/1510981/shortest-distance-between-two-general-curves-using-matlab – NoChance Sep 13 '19 at 21:34
  • We seem to have a language problem here. We do not have two lines, and two lines in the plane can never be skew. You should rewrite the question accurately: You want the distance between a line and a curve in the plane. – Ted Shifrin Sep 13 '19 at 22:12
  • "2 skewed lines in 2D" doesn't properly describe your setup. You have one line and one curve (a parabola) which isn't a line. If you did have two skewed lines in 2D, then they would intersect somewhere and thus have zero distance between them. – hardmath Sep 13 '19 at 22:14

3 Answers3

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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,

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

Quanto
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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.

Sheridan Grant
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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 $x=-1/4$.

Michael Hoppe
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