# Thread: Finding minimum distance between two points on two different parabolas

1. ## Finding minimum distance between two points on two different parabolas

Hi everyone. I've got a homework problem with Differential Calculus that drives me crazy for a couple of days so far and would like to ask you guys for a few suggestions

"Given two parabolas, (C1): y = x^2 and (C2): y = - (4-x)^2. Find two points, each point on one parabola so that the distance between them is smallest."

What I got so far: Well, I called the point on (C1) (x1, y1), the other point (x2, y2). After that, I set up the distance equation and replace both y(s) with their respective x(s) as follow:

After that I got stuck ... since I don't know how to differentiate an equation with two variables. Here I need to find the minimum of d, which is the distance between the two points.
I wonder if my approach to the problem is wrong or what. Any suggestion is highly appreciated. Thank you. (Also sorry if I confused you guys because I don't know how to type subscript and superscript in the forum yet)

2. Pls find attached work.

3. Originally Posted by Veronica1999
Pls find attached work.
Thank you for your answer, but would you please explain a little bit about what you did? I don't get why taking the absolute value of (y2- y1). I thought we need to use the distance formula... thanks

4. Pls find attached work. Hope it is helpful.

5. OK I understand what you wrote in the latest attachment. However, in this case, the two parabolas do not appear on straight line. One is x^2 and has its vertex at (0,0) and concave upward, while the other has a vertex at (4,0) and looks downward... so somehow I don't think the distance between the two points is |y2 - y1| ...

I just don't get it at this point. Please bear with me as I'm just in my first semester calculus. Thank you.

6. Originally Posted by sparkyboy
Hi everyone. I've got a homework problem with Differential Calculus that drives me crazy for a couple of days so far and would like to ask you guys for a few suggestions

"Given two parabolas, (C1): y = x^2 and (C2): y = - (4-x)^2. Find two points, each point on one parabola so that the distance between them is smallest."

What I got so far: Well, I called the point on (C1) (x1, y1), the other point (x2, y2). After that, I set up the distance equation and replace both y(s) with their respective x(s) as follow:

After that I got stuck ... since I don't know how to differentiate an equation with two variables. Here I need to find the minimum of d, which is the distance between the two points.
I wonder if my approach to the problem is wrong or what. Any suggestion is highly appreciated. Thank you. (Also sorry if I confused you guys because I don't know how to type subscript and superscript in the forum yet)
the common normal is shortest(think why this is so?). its easy to find the common normal if you know the parametric form of the normal to a parabola.

7. Originally Posted by abhishekkgp
the common normal is shortest(think why this is so?). its easy to find the common normal if you know the parametric form of the normal to a parabola.
Umm... sorry once again but I don't know what is a common normal. I'm just in my first semester calculus ...

8. Originally Posted by sparkyboy
Hi everyone. I've got a homework problem with Differential Calculus that drives me crazy for a couple of days so far and would like to ask you guys for a few suggestions

"Given two parabolas, (C1): y = x^2 and (C2): y = - (4-x)^2. Find two points, each point on one parabola so that the distance between them is smallest."

What I got so far: Well, I called the point on (C1) (x1, y1), the other point (x2, y2). After that, I set up the distance equation and replace both y(s) with their respective x(s) as follow:

After that I got stuck ... since I don't know how to differentiate an equation with two variables. Here I need to find the minimum of d, which is the distance between the two points.
I wonder if my approach to the problem is wrong or what. Any suggestion is highly appreciated. Thank you. (Also sorry if I confused you guys because I don't know how to type subscript and superscript in the forum yet)

to type subscript use '_', that is "x subscript n" will be x_n. for superscript use "^".

9. Originally Posted by abhishekkgp
the common normal is shortest(think why this is so?). its easy to find the common normal if you know the parametric form of the normal to a parabola.
there is one more way to go about this. chose a point $\displaystyle (x_1,y_1)$ on the first parabola and fix it. Now try to solve the following one-variable-calculus problem.
Find a point on the second parabola $\displaystyle (x_2,y_2)$ such that the distance between theses two points is minimum.

Note that $\displaystyle x_1$ is a constant now. you have to find $\displaystyle x_2$ in terms of $\displaystyle x_1$.

10. Originally Posted by sparkyboy
Umm... sorry once again but I don't know what is a common normal. I'm just in my first semester calculus ...
A normal at a point is a line perpendicular to the tangent at that point. a common normal is a line normal to two curves in question at the same time.

11. Originally Posted by abhishekkgp
there is one more way to go about this. chose a point $\displaystyle (x_1,y_1)$ on the first parabola and fix it. Now try to solve the following one-variable-calculus problem.
Find a point on the second parabola $\displaystyle (x_2,y_2)$ such that the distance between theses two points is minimum.

Note that $\displaystyle x_1$ is a constant now. you have to find $\displaystyle x_2$ in terms of $\displaystyle x_1$.
So you mean to turn the problem into "find the shortest distance from a point to a curve". I can do it. But I don't get why doing that? I need two points though ...

12. Originally Posted by sparkyboy
So you mean to turn the problem into "find the shortest distance from a point to a curve". I can do it. But I don't get why doing that? I need two points though ...
YES, "first find the shortest distance from a point to a curve". this expression of shortest distance will be completely the function of $\displaystyle x_1$. minimize this expression too! and that will be the answer. (do you see why?)

13. [QUOTE=sparkyboy;647734]OK I understand what you wrote in the latest attachment. However, in this case, the two parabolas do not appear on straight line. One is x^2 and has its vertex at (0,0) and concave upward, while the other has a vertex at (4,0) and looks downward... so somehow I don't think the distance between the two points is |y2 - y1| ...

I just don't get it at this point. Please bear with me as I'm just in my first semester calculus. Thank you.[/QUOTE

Pls find attached graph.

14. YES, "first find the shortest distance from a point to a curve". this expression of shortest distance will be completely the function of x_1. minimize this expression too! and that will be the answer. (do you see why?)
Umm ... I still don't understand, would you please tell me why? Thank you

Originally Posted by sparkyboy
OK I understand what you wrote in the latest attachment. However, in this case, the two parabolas do not appear on straight line. One is x^2 and has its vertex at (0,0) and concave upward, while the other has a vertex at (4,0) and looks downward... so somehow I don't think the distance between the two points is |y2 - y1| ...

I just don't get it at this point. Please bear with me as I'm just in my first semester calculus. Thank you.

Pls find attached graph.
Thank you for the graph. However, I'm still confused. What if the first point is (-2, 4) and the second point is (3, -1)? Then the distance cannot be a straight line right? I mean I still don't get why the distance is just simply |y2 - y1|. Really sorry about my dumbness but please bear with me. Thank you.

15. Originally Posted by sparkyboy
Umm ... I still don't understand, would you please tell me why? Thank you
you must be aware of the identity min{a,b,c,d}=min{min{a,b},min{c,d}}. what i told you to do is more or less the same. we first find minimum distance at different values of $\displaystyle x_1$, then we take the minimum one among these.

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### shortest distanc between parabola nd parabola without using differentiation

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