Dgdrxrt

Can I legally use front facing blue light in the UK?

"listening to me about as much as you're listening to this pole here"

Calculate Levenshtein distance between two strings in Python

What are the advantages and disadvantages of running one shots compared to campaigns?

Why is the design of haulage companies so “special”?

What to wear for invited talk in Canada

Latin words with no plurals in English

Need help identifying/translating a plaque in Tangier, Morocco

Is every set a filtered colimit of finite sets?

COUNT(id) or MAX(id) - which is faster?

Patience, young "Padovan"

Why is my log file so massive? 22gb. I am running log backups

Can I find out the caloric content of bread by dehydrating it?

Manga about a female worker who got dragged into another world together with this high school girl and she was just told she's not needed anymore

Could Giant Ground Sloths have been a Good Pack Animal for the Ancient Mayans

"My colleague's body is amazing"

Symmetry in quantum mechanics

Extreme, but not acceptable situation and I can't start the work tomorrow morning

Why doesn't a const reference extend the life of a temporary object passed via a function?

Why do UK politicians seemingly ignore opinion polls on Brexit?

How can I fix this gap between bookcases I made?

aging parents with no investments

Prime joint compound before latex paint?

What is the meaning of "of trouble" in the following sentence?

Minimum value of length of tangent of the ellipse $x^2/a^2 + y^2/b^2 = 1$, intercepted between the co-ordinate axes

find the center of an ellipse given tangent point and anglePassing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]Equation of hyperbolaGeometric proof of this property of the ellipseQuestion related to elliptical anglesFinding the axes of an ellipse from deformation of a circleDetermine angular coordinate of contact point between a rotated ellipse and its tangentFind the angle between the two tangents drawn from the point $(1,2)$ to the ellipse $x^2+2y^2=3$.Finding the major and minor axes lengths of an ellipse given parametric equationsA tangent to an ellipse makes angles $alpha$ with major axis and $beta$ with a focal radius; show that the eccentricity is $cosbeta/cosalpha$.

0

$begingroup$

I have taken a parameter $(a cos c, b sin c)$ where $c$ is the eccentric angle and the tangent passing through this point cuts the x-axis at the point $(a cos c, 0)$ and y-axis at $(0,b sin c)$.

After this I have calculated the the length using Pythagoras theorem. But I couldn't get the minimum value.

conic-sections

share|cite|improve this question

edited Apr 1 at 3:39

Ertxiem

671112

asked Mar 30 at 2:55

Ayus DasAyus Das

32

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Can you please clarify what you mean by minimum length? Are you talking about the ellipse minor axis?
  $endgroup$
  – Ertxiem
  Apr 1 at 3:10
  

  
  
  
  

add a comment | 

0

$begingroup$

I have taken a parameter $(a cos c, b sin c)$ where $c$ is the eccentric angle and the tangent passing through this point cuts the x-axis at the point $(a cos c, 0)$ and y-axis at $(0,b sin c)$.

After this I have calculated the the length using Pythagoras theorem. But I couldn't get the minimum value.

conic-sections

share|cite|improve this question

edited Apr 1 at 3:39

Ertxiem

671112

asked Mar 30 at 2:55

Ayus DasAyus Das

32

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Can you please clarify what you mean by minimum length? Are you talking about the ellipse minor axis?
  $endgroup$
  – Ertxiem
  Apr 1 at 3:10
  

  
  
  
  

add a comment | 

$begingroup$

I have taken a parameter $(a cos c, b sin c)$ where $c$ is the eccentric angle and the tangent passing through this point cuts the x-axis at the point $(a cos c, 0)$ and y-axis at $(0,b sin c)$.

After this I have calculated the the length using Pythagoras theorem. But I couldn't get the minimum value.

conic-sections

share|cite|improve this question

edited Apr 1 at 3:39

Ertxiem

671112

asked Mar 30 at 2:55

Ayus DasAyus Das

32

$endgroup$

share|cite|improve this question

edited Apr 1 at 3:39

Ertxiem

671112

asked Mar 30 at 2:55

Ayus DasAyus Das

32

share|cite|improve this question

edited Apr 1 at 3:39

Ertxiem

671112

asked Mar 30 at 2:55

Ayus DasAyus Das

32

edited Apr 1 at 3:39

Ertxiem

671112

edited Apr 1 at 3:39

Ertxiem

671112

asked Mar 30 at 2:55

Ayus DasAyus Das

32

asked Mar 30 at 2:55

Ayus DasAyus Das

32

1 Answer
1

active

oldest

votes

1

$begingroup$

Length of tangent intercepted:
$sqrt frac a^2 cos^2 t+ frac b^2 sin^2t$
= $sqrt a^2 sec^2 t+ b^2 cosec^2t$
= $sqrt a^2tan^2t+a^2+ b^2 cot^2t+b^2$
Using AM-GM inequality:
$frac a^2tan^2t+b^2 cot^2t 2 ge sqrt a^2tan^2t.b^2 cot^2t$
So $a^2tan^2t+b^2 cot^2t ge 2ab $
Therefore required minimum value is $a+b$

share|cite|improve this answer

edited Mar 30 at 3:14

answered Mar 30 at 3:08

TojrahTojrah

4036

$endgroup$

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
);



);

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3167873%2fminimum-value-of-length-of-tangent-of-the-ellipse-x2-a2-y2-b2-1-inter%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

1

$begingroup$

Length of tangent intercepted:
$sqrt frac a^2 cos^2 t+ frac b^2 sin^2t$
= $sqrt a^2 sec^2 t+ b^2 cosec^2t$
= $sqrt a^2tan^2t+a^2+ b^2 cot^2t+b^2$
Using AM-GM inequality:
$frac a^2tan^2t+b^2 cot^2t 2 ge sqrt a^2tan^2t.b^2 cot^2t$
So $a^2tan^2t+b^2 cot^2t ge 2ab $
Therefore required minimum value is $a+b$

share|cite|improve this answer

edited Mar 30 at 3:14

answered Mar 30 at 3:08

TojrahTojrah

4036

$endgroup$

add a comment | 

1

$begingroup$

Length of tangent intercepted:
$sqrt frac a^2 cos^2 t+ frac b^2 sin^2t$
= $sqrt a^2 sec^2 t+ b^2 cosec^2t$
= $sqrt a^2tan^2t+a^2+ b^2 cot^2t+b^2$
Using AM-GM inequality:
$frac a^2tan^2t+b^2 cot^2t 2 ge sqrt a^2tan^2t.b^2 cot^2t$
So $a^2tan^2t+b^2 cot^2t ge 2ab $
Therefore required minimum value is $a+b$

share|cite|improve this answer

edited Mar 30 at 3:14

answered Mar 30 at 3:08

TojrahTojrah

4036

$endgroup$

add a comment | 

$begingroup$

Length of tangent intercepted:
$sqrt frac a^2 cos^2 t+ frac b^2 sin^2t$
= $sqrt a^2 sec^2 t+ b^2 cosec^2t$
= $sqrt a^2tan^2t+a^2+ b^2 cot^2t+b^2$
Using AM-GM inequality:
$frac a^2tan^2t+b^2 cot^2t 2 ge sqrt a^2tan^2t.b^2 cot^2t$
So $a^2tan^2t+b^2 cot^2t ge 2ab $
Therefore required minimum value is $a+b$

share|cite|improve this answer

edited Mar 30 at 3:14

answered Mar 30 at 3:08

TojrahTojrah

4036

$endgroup$

share|cite|improve this answer

edited Mar 30 at 3:14

answered Mar 30 at 3:08

TojrahTojrah

4036

answered Mar 30 at 3:08

TojrahTojrah

4036

answered Mar 30 at 3:08

TojrahTojrah

4036