# Find the minimum distance between the curves

1. Sep 13, 2009

### utsav55

1. The problem statement, all variables and given/known data
Find the minimum distance between the curves (Parabola) y^2 = x-1 and x^2 = y-1

2. Relevant equations
y^2 = x-1
x^2 = y-1

3. The attempt at a solution
Tried to find the distance between their vertex, but the answer was wrong and no where near.

2. Sep 13, 2009

### Hootenanny

Staff Emeritus
Welcome to Physics Forums.
In general, what is the distance between two points?

3. Sep 13, 2009

### utsav55

Well, we can use the distance formula to find the distance between 2 points, but in this question between which 2 points we need to find the distance and why?

4. Sep 13, 2009

### Hootenanny

Staff Emeritus
The distance formula is used to compute the distance between two sets of points, say (x1, y1) and (x2, y2). In general, these can be any two points. However, in this case we know that these points must lie on two curves. So we can replace one co-ordinate in each case with an expression in terms of the remaining co-ordinate.

Do you follow?

5. Sep 13, 2009

### utsav55

Just didn't got the last sentence.

6. Sep 13, 2009

### Hootenanny

Staff Emeritus
Instead of finding the distance between the two points (x1,y1) and (x2,y2) you want to find the distance between, say,

$$\left({y_1}^2+1,y_1\right)\text{ and } \left(x_2, {x_2}^2+1\right)$$

Does that make sense?

7. Sep 13, 2009

### utsav55

Please explain me that how you arrived at that conclusion, sorry I didn't got that...

8. Sep 13, 2009

### Hootenanny

Staff Emeritus
Okay let us take a point (x1, y1) on the curve y2 = x-1, and another point, (x2, y2) on the curve x2 = y-1. Now, suppose that these two points represent the closest points on the two curves. We want to find the distance between them. Currently, we have four variables: x1, x2, y1 and y2; but we can reduce the number of variables. Since we know that the points must lie on their respective curves we can simply substitute one co-ordinate into the equation of the curve. For example, let us take the first point: (x1, y1). We know that since this point lies on the first curve it must satisfy the equation y2 = x-1. In other words, y12 = x1-1. Hence, x1 = y12+1. Therefore we can re-write the point (x1, y 1) in terms of y1 only: (y12+1, y1).

Do you now follow?

9. Sep 13, 2009

### utsav55

Yes, got till here.
How to proceed from here please? We got 2 points, 1 lying on 1 curve. How do we find the distance now?
The answer is a numeric value...

Last edited: Sep 13, 2009
10. Sep 13, 2009

### Hootenanny

Staff Emeritus
The next step would be to repeat the above steps for the second point, yielding the second expression I stated in my previous post. Then, we have the distance formula

$$d = \sqrt{\left(x_2 - x_1\right)^2 - \left(y_2-y_1\right)^2}$$

Now, we substitute in our two points,

$$d = \sqrt{\left(x_2 - {y_1}^2-1\right)^2 - \left({x_2}^2-y_1 +1 \right)^2}$$

So, you want to find the minimum distance between the two curves.

What do you think out next step would be?

11. Sep 13, 2009

### utsav55

Use maxima/minima concept??

12. Sep 13, 2009

### Hootenanny

Staff Emeritus
Indeed. So you want to minimise d(x2, y1) with respect to x2 and y1. It would be useful to note that,

$$d\left(x_2,y_1\right) = \sqrt{f\left(x_2,y_1\right)}$$

Hence, one could simply minimise f in order to find the minimum of d.