Mixing
Find symmetric points with respect to the unit circle
I have struggled with an exercise, namely: find the set of symmetric points with respect to the unit circle of a circle given by this equation: $ |z-1|=1$, I have an idea of what this might be. Since points, A and B, which are the intersection points of both circles will not change when taking symmetry and also point 2 will go to 1/2. So I guess that the set will be the line crossing all 3 points A, B and 1/2, but I would like some explanation added to this. Thanks, any help appreciated.
complex-numbers complex-geometry
asked Nov 21 '18 at 16:11
ryszard eggink
308110
add a comment |
I have struggled with an exercise, namely: find the set of symmetric points with respect to the unit circle of a circle given by this equation: $ |z-1|=1$, I have an idea of what this might be. Since points, A and B, which are the intersection points of both circles will not change when taking symmetry and also point 2 will go to 1/2. So I guess that the set will be the line crossing all 3 points A, B and 1/2, but I would like some explanation added to this. Thanks, any help appreciated.
complex-numbers complex-geometry
asked Nov 21 '18 at 16:11
ryszard eggink
308110
add a comment |
I have struggled with an exercise, namely: find the set of symmetric points with respect to the unit circle of a circle given by this equation: $ |z-1|=1$, I have an idea of what this might be. Since points, A and B, which are the intersection points of both circles will not change when taking symmetry and also point 2 will go to 1/2. So I guess that the set will be the line crossing all 3 points A, B and 1/2, but I would like some explanation added to this. Thanks, any help appreciated.
complex-numbers complex-geometry
asked Nov 21 '18 at 16:11
ryszard eggink
308110
I have struggled with an exercise, namely: find the set of symmetric points with respect to the unit circle of a circle given by this equation: $ |z-1|=1$, I have an idea of what this might be. Since points, A and B, which are the intersection points of both circles will not change when taking symmetry and also point 2 will go to 1/2. So I guess that the set will be the line crossing all 3 points A, B and 1/2, but I would like some explanation added to this. Thanks, any help appreciated.
complex-numbers complex-geometry
complex-numbers complex-geometry
asked Nov 21 '18 at 16:11
ryszard eggink
308110
asked Nov 21 '18 at 16:11
ryszard eggink
308110
asked Nov 21 '18 at 16:11
ryszard eggink
308110
asked Nov 21 '18 at 16:11
ryszard eggink
308110
asked Nov 21 '18 at 16:11
ryszard eggink
308110
308110
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Hint: Let $z$ and $w$ be on circle $|z-1|=1$, then
$$|z-1|=1~~~~;~~~~|w-1|=1$$
these points are symmetric respct to the unit circle, means there is a $thetainmathbb R$ such that
$$dfrac{z+w}{2}=e^{itheta}$$
or $|z+w|=2$, therefore with deleting $w$ among equations
$$|z-1|=1~~~~;~~~~|w-1|=1~~~~;~~~~|z+w|=2$$
we can find desired points $z$.
answered Nov 21 '18 at 19:07
Nosrati
26.5k62354
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%2f3007934%2ffind-symmetric-points-with-respect-to-the-unit-circle%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
Hint: Let $z$ and $w$ be on circle $|z-1|=1$, then
$$|z-1|=1~~~~;~~~~|w-1|=1$$
these points are symmetric respct to the unit circle, means there is a $thetainmathbb R$ such that
$$dfrac{z+w}{2}=e^{itheta}$$
or $|z+w|=2$, therefore with deleting $w$ among equations
$$|z-1|=1~~~~;~~~~|w-1|=1~~~~;~~~~|z+w|=2$$
we can find desired points $z$.
answered Nov 21 '18 at 19:07
Nosrati
26.5k62354
add a comment |
Hint: Let $z$ and $w$ be on circle $|z-1|=1$, then
$$|z-1|=1~~~~;~~~~|w-1|=1$$
these points are symmetric respct to the unit circle, means there is a $thetainmathbb R$ such that
$$dfrac{z+w}{2}=e^{itheta}$$
or $|z+w|=2$, therefore with deleting $w$ among equations
$$|z-1|=1~~~~;~~~~|w-1|=1~~~~;~~~~|z+w|=2$$
we can find desired points $z$.
answered Nov 21 '18 at 19:07
Nosrati
26.5k62354
add a comment |
Hint: Let $z$ and $w$ be on circle $|z-1|=1$, then
$$|z-1|=1~~~~;~~~~|w-1|=1$$
these points are symmetric respct to the unit circle, means there is a $thetainmathbb R$ such that
$$dfrac{z+w}{2}=e^{itheta}$$
or $|z+w|=2$, therefore with deleting $w$ among equations
$$|z-1|=1~~~~;~~~~|w-1|=1~~~~;~~~~|z+w|=2$$
we can find desired points $z$.
answered Nov 21 '18 at 19:07
Nosrati
26.5k62354
Hint: Let $z$ and $w$ be on circle $|z-1|=1$, then
$$|z-1|=1~~~~;~~~~|w-1|=1$$
these points are symmetric respct to the unit circle, means there is a $thetainmathbb R$ such that
$$dfrac{z+w}{2}=e^{itheta}$$
or $|z+w|=2$, therefore with deleting $w$ among equations
$$|z-1|=1~~~~;~~~~|w-1|=1~~~~;~~~~|z+w|=2$$
we can find desired points $z$.
answered Nov 21 '18 at 19:07
Nosrati
26.5k62354
edited Nov 21 '18 at 19:14
answered Nov 21 '18 at 19:07
Nosrati
26.5k62354
answered Nov 21 '18 at 19:07
Nosrati
26.5k62354
answered Nov 21 '18 at 19:07
Nosrati
26.5k62354
26.5k62354
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3007934%2ffind-symmetric-points-with-respect-to-the-unit-circle%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
