circle inside an ellipse with fixed width but variable length












0












$begingroup$


image of the diagram



Inside an ellipse with width 'a' and length 'b', there is a circle. The circle touches the point (-a,0), which means circle touches the edge of the ellipse but ellipse doesn't cut through the circle. I need to find the minimum value or the range of 'b' when the radius of the circle is 'r' and 'a'='R'. How do I do it? note: a_0=R-r










share|cite|improve this question











$endgroup$












  • $begingroup$
    If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
    $endgroup$
    – David K
    Jan 10 at 16:20










  • $begingroup$
    Hint: Show that the center of the circle lies on a focus of the ellipse.
    $endgroup$
    – amd
    Jan 11 at 2:16
















0












$begingroup$


image of the diagram



Inside an ellipse with width 'a' and length 'b', there is a circle. The circle touches the point (-a,0), which means circle touches the edge of the ellipse but ellipse doesn't cut through the circle. I need to find the minimum value or the range of 'b' when the radius of the circle is 'r' and 'a'='R'. How do I do it? note: a_0=R-r










share|cite|improve this question











$endgroup$












  • $begingroup$
    If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
    $endgroup$
    – David K
    Jan 10 at 16:20










  • $begingroup$
    Hint: Show that the center of the circle lies on a focus of the ellipse.
    $endgroup$
    – amd
    Jan 11 at 2:16














0












0








0


1



$begingroup$


image of the diagram



Inside an ellipse with width 'a' and length 'b', there is a circle. The circle touches the point (-a,0), which means circle touches the edge of the ellipse but ellipse doesn't cut through the circle. I need to find the minimum value or the range of 'b' when the radius of the circle is 'r' and 'a'='R'. How do I do it? note: a_0=R-r










share|cite|improve this question











$endgroup$




image of the diagram



Inside an ellipse with width 'a' and length 'b', there is a circle. The circle touches the point (-a,0), which means circle touches the edge of the ellipse but ellipse doesn't cut through the circle. I need to find the minimum value or the range of 'b' when the radius of the circle is 'r' and 'a'='R'. How do I do it? note: a_0=R-r







geometry functions






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 10 at 10:55







nickname

















asked Jan 10 at 10:53









nicknamenickname

1




1












  • $begingroup$
    If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
    $endgroup$
    – David K
    Jan 10 at 16:20










  • $begingroup$
    Hint: Show that the center of the circle lies on a focus of the ellipse.
    $endgroup$
    – amd
    Jan 11 at 2:16


















  • $begingroup$
    If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
    $endgroup$
    – David K
    Jan 10 at 16:20










  • $begingroup$
    Hint: Show that the center of the circle lies on a focus of the ellipse.
    $endgroup$
    – amd
    Jan 11 at 2:16
















$begingroup$
If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
$endgroup$
– David K
Jan 10 at 16:20




$begingroup$
If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
$endgroup$
– David K
Jan 10 at 16:20












$begingroup$
Hint: Show that the center of the circle lies on a focus of the ellipse.
$endgroup$
– amd
Jan 11 at 2:16




$begingroup$
Hint: Show that the center of the circle lies on a focus of the ellipse.
$endgroup$
– amd
Jan 11 at 2:16










1 Answer
1






active

oldest

votes


















1












$begingroup$

The smallest value of $b$ is that leading to tangency between ellipse and circle.



Write down the equations of the ellipse and circle, eliminate $y^2$ to get a single quadratic equation in $x$. Ellipse and circle are tangent when that equation has a single solution, i.e. when its discriminant vanishes.






share|cite|improve this answer









$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3068491%2fcircle-inside-an-ellipse-with-fixed-width-but-variable-length%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    The smallest value of $b$ is that leading to tangency between ellipse and circle.



    Write down the equations of the ellipse and circle, eliminate $y^2$ to get a single quadratic equation in $x$. Ellipse and circle are tangent when that equation has a single solution, i.e. when its discriminant vanishes.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      The smallest value of $b$ is that leading to tangency between ellipse and circle.



      Write down the equations of the ellipse and circle, eliminate $y^2$ to get a single quadratic equation in $x$. Ellipse and circle are tangent when that equation has a single solution, i.e. when its discriminant vanishes.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        The smallest value of $b$ is that leading to tangency between ellipse and circle.



        Write down the equations of the ellipse and circle, eliminate $y^2$ to get a single quadratic equation in $x$. Ellipse and circle are tangent when that equation has a single solution, i.e. when its discriminant vanishes.






        share|cite|improve this answer









        $endgroup$



        The smallest value of $b$ is that leading to tangency between ellipse and circle.



        Write down the equations of the ellipse and circle, eliminate $y^2$ to get a single quadratic equation in $x$. Ellipse and circle are tangent when that equation has a single solution, i.e. when its discriminant vanishes.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 10 at 14:14









        AretinoAretino

        23.5k21443




        23.5k21443






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3068491%2fcircle-inside-an-ellipse-with-fixed-width-but-variable-length%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            Popular posts from this blog

            android studio warns about leanback feature tag usage required on manifest while using Unity exported app?

            SQL update select statement

            'app-layout' is not a known element: how to share Component with different Modules