Find the minimum distance between the curves

[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.

[Hootenanny]

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

[utsav55]

Welcome to Physics Forums.

In general, what is the distance between two points?

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?

[Hootenanny]

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?
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?

[utsav55]

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?

Just didn't got the last sentence.

[Hootenanny]

Just didn't got the last sentence.
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?

[utsav55]

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?

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

[Hootenanny]

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

[utsav55]

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?

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...

[Hootenanny]

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...
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?

[utsav55]

Use maxima/minima concept??

[Hootenanny]

Use maxima/minima concept??
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.