The discussion focuses on finding the minimum distance between the curves defined by the equations \(y^2 = x - 1\) and \(x^2 = y - 1\). Participants explore the use of the distance formula to compute the distance between points on these curves, specifically substituting coordinates to express points in terms of a single variable. The final approach involves minimizing the distance function \(d(x_2, y_1) = \sqrt{f(x_2, y_1)}\) using calculus techniques to find the minimum distance between the two parabolas.
PREREQUISITESStudents and educators in mathematics, particularly those studying calculus and analytical geometry, as well as anyone interested in optimization problems involving curves.
In general, what is the distance between two points?utsav55 said:Homework Statement
Find the minimum distance between the curves (Parabola) y^2 = x-1 and x^2 = y-1Homework Equations
y^2 = x-1
x^2 = y-1The Attempt at a Solution
Tried to find the distance between their vertex, but the answer was wrong and no where near.
Hootenanny said:Welcome to Physics Forums.
In general, what is the distance between two points?
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.utsav55 said: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 said: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?
Instead of finding the distance between the two points (x1,y1) and (x2,y2) you want to find the distance between, say,utsav55 said:Just didn't got the last sentence.
Hootenanny said: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?
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).utsav55 said:Please explain me that how you arrived at that conclusion, sorry I didn't got that...
Hootenanny said: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?
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 formulautsav55 said: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...
Indeed. So you want to minimise d(x2, y1) with respect to x2 and y1. It would be useful to note that,utsav55 said:Use maxima/minima concept??