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Math Help - Finding minimum distance between two points on two different parabolas

  1. #1
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    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)
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    Member Veronica1999's Avatar
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    Pls find attached work.
    Attached Files Attached Files
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    Quote Originally Posted by Veronica1999 View Post
    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
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    Pls find attached work. Hope it is helpful.
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    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.
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    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by sparkyboy View Post
    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.
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    Quote Originally Posted by abhishekkgp View Post
    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 ...
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    Quote Originally Posted by sparkyboy View Post
    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 "^".
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    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by abhishekkgp View Post
    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 (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 (x_2,y_2) such that the distance between theses two points is minimum.

    Note that x_1 is a constant now. you have to find x_2 in terms of x_1.
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  10. #10
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    Quote Originally Posted by sparkyboy View Post
    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.
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    Quote Originally Posted by abhishekkgp View Post
    there is one more way to go about this. chose a point (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 (x_2,y_2) such that the distance between theses two points is minimum.

    Note that x_1 is a constant now. you have to find x_2 in terms of 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 ...
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    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by sparkyboy View Post
    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 x_1. minimize this expression too! and that will be the answer. (do you see why?)
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    Member Veronica1999's Avatar
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    [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.
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    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

    Quote Originally Posted by sparkyboy View Post
    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.
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  15. #15
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by sparkyboy View Post
    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 x_1, then we take the minimum one among these.
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