Find the minimum distance between the curves

[utsav55]

Homework Statement

Find the minimum distance between the curves (Parabola) y^2 = x-1 and x^2 = y-1

Homework Equations

y^2 = x-1
x^2 = y-1

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.

Homework Statement

Find the minimum distance between the curves (Parabola) y^2 = x-1 and x^2 = y-1

Homework Equations

y^2 = x-1
x^2 = y-1

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.