circle inside an ellipse with fixed width but variable length
$begingroup$
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
$endgroup$
add a comment |
$begingroup$
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
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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
$endgroup$
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
geometry functions
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
add a comment |
$begingroup$
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
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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.
$endgroup$
add a comment |
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
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.
answered Jan 10 at 14:14
AretinoAretino
23.5k21443
23.5k21443
add a comment |
add a comment |
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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
$begingroup$
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