**An ellipse is inscribed in a semi circle touches the circular arc at two distinct points and also touches the bounding diameter. its major axis is parallel to the bounding diameter, When the ellipse has maximum possible area, its eccentricity is?**

How do you solve this question, this came in the **KVPY** exam that was held on 2nd november, and i couldnt solve this problem,
though i did guess it by simply finding which **e** gave the largest area,

Can any one give the actual solution

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TopNewestMy \(e\) is coming out to be \(\sqrt{\frac{2}{3}}\). Kindly tell me whether it is correct or not. If I am correct I will post the solution. – Ronak Agarwal · 2 years, 6 months ago

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@Ronak Agarwal – Karan Shekhawat · 2 years, 6 months ago

will you please upload Your SolutionLog in to reply

Image

Let the point of intersection as shown in the figure be \((acos(t),b(1+sin(t)))\)

With respect to circle the point be \(A=(rcos(\theta),rsin(\theta))\)

Since the points coincide hence :

\(rcos(\theta)=acos(t)\) (i)

\(rsin(\theta)=b(sin(t)+1)\) (ii)

Dividing them we get :

\(tan(\theta)=\frac{b(1+sin(t))}{acos(t)}\) (iii)

We know that the tangents of the circle and ellipse at that point coincide.

Hence their slopes are equal.

\({m}_{ellipse}={m}_{circle}\)

\(\Rightarrow \frac{-bcot(t)}{a}=-cot(\theta)\) (iv)

Multiplying (iii) and (iv)

\(1=\frac{{b}^{2}(1+sin(t))}{{a}^{2}sin(t)}\) (v)

Squaring and adding (i) and (ii) :

\({r}^{2}={a}^{2}{cos}^{2}(t)+{b}^{2}{(1+sin(t))}^{2}\)

Using (v) we get :

\({a}^{2}{cos}^{2}(t)+{a}^{2}sin(t)(1+sin(t))={r}^{2}\)

\(\Rightarrow {a}^{2}=\frac{{r}^{2}}{1+sin(t)}\)

Hence \({b}^{2}=\frac{{r}^{2}sin(t)}{{1+sin(t)}^{2}}\)

So area of ellipse =\(A=\pi ab = {r}^{2} \sqrt{\frac{sin(t)}{{(1+sin(t)}^{3}}}\)

Maximising this we get the maximum at \(sin(t)=\frac{1}{2}\)

Hence we get :

\(\frac{{b}^{2}}{{a}^{2}}=\frac{1}{3}\) (Using (v))

\(\Rightarrow e=\sqrt{\frac{2}{3}}\) – Ronak Agarwal · 2 years, 6 months ago

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– Mvs Saketh · 2 years, 6 months ago

That was awesome.... simply awesome... thanks for uploading...Log in to reply

@Ronak Agarwal You did excellent work ! I used Co-ordinate geometry which is really too bad method in front of you :)

oh greatI used \(C:\quad { x }^{ 2 }\quad +\quad { (y+b) }^{ 2 }=\quad { R }^{ 2 }\\ \quad \\ E:\quad \cfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\cfrac { { \quad y }^{ 2 } }{ { b }^{ 2 } } \quad =\quad 1\).

And further which needs at-least 2 pages which is useless in front of Yours :) – Deepanshu Gupta · 2 years, 6 months ago

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@Ronak Agarwal But Ronak shoudn't it is b(sin(t)) instead of b(1+ sin(t) ) Plz Explain me ! Thanks – Karan Shekhawat · 2 years, 6 months ago

Great !!! Thanks a lot !Log in to reply

– Ronak Agarwal · 2 years, 6 months ago

Actually with respect to origin the ellipse has it's centre at (0,b) hence I have shifted it's co-ordinates accordingly.Log in to reply

– Karan Shekhawat · 2 years, 6 months ago

Thanks ! Now I got it completely ! I really appreciate your solution very much !!Log in to reply

– Ronak Agarwal · 2 years, 6 months ago

ThanksLog in to reply

– Ronak Agarwal · 2 years, 6 months ago

Okay uploading it.Log in to reply

How much marks are you getting @Mvs Saketh . I'm just asking. – Ronak Agarwal · 2 years, 6 months ago

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– Mvs Saketh · 2 years, 6 months ago

also if a question turns out to be wrong, then will everyone be given marks? ( i am asking since you are already a kvpy scholar)Log in to reply

– Ayush Garg · 2 years, 6 months ago

Hey! which question are you talking about??Log in to reply

– Ronak Agarwal · 2 years, 6 months ago

Maths Part-2, you can see this question there.Log in to reply

what about you? @Ronak Agarwal – Mvs Saketh · 2 years, 6 months ago

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– Ronak Agarwal · 2 years, 6 months ago

I am already a KVPY scholar hence I haven't given KVPY this year.Log in to reply

– Mvs Saketh · 2 years, 6 months ago

Oh, ok so bro can you please tell me what are the chances of getting through 2nd level for me(based on score)? and is there hope for me to get into IISC,, if not through this,, will qualifying NSEP (if i am able to) help?Log in to reply

– Ronak Agarwal · 2 years, 6 months ago

Your are sure to go to 2nd stage but you have to perform a lot better in the interview, also can you please tell me what are the benifits of going into IISC as I am also interested in taking admission into this institution, but I am very doubtful about the oppurtunities after my graduation. I am very much interested in going into research field.Log in to reply

And if u want to spend the rest of your life wondering,, i think its awesome ,,, – Mvs Saketh · 2 years, 6 months ago

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– Ronak Agarwal · 2 years, 6 months ago

I am very confused whether to go for BTech or BSc MSc.Log in to reply

– Mvs Saketh · 2 years, 6 months ago

I think u will excel in whatever u do!Log in to reply

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@Mvs Saketh !! what is procedure to get into IISC if a person never given the K.V.P.Y ?? Does JEE advance Helps in it ? If It is Then What is restrictions in ranks in it for getting an IISC ??

HeyAnd Also in this above Question Is Radius is fixed or not ? I can't understand what does this question want to convey will you please clarify it to me ? – Deepanshu Gupta · 2 years, 6 months ago

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– Krishna Sharma · 2 years, 6 months ago

Do you know that when IISC started( somewhat bettter than IIT ) rarely people knew about it and the students who filled the form (very few)got admission directly :PLog in to reply

@DEEPANSHU GUPTA yes JEE advance does help,, minimum is above 60 percent,, but thats just the minimum, i heard that it is recommended to get a within 1000 rank,, and as far as i know olympiads help in it, which is why i am highly interested in NSEP,,

Radius is fixed,, its value has not been given, andyeah question says of all ellipses that can be inscribed in a semi circle , the ellipse has maximum area for what value of e (eccentricity) – Mvs Saketh · 2 years, 6 months ago

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– Deepanshu Gupta · 2 years, 6 months ago

Ok I got it ... Let me try !! And sorry but is any olyimpiad at 23rd Nov. ?? or something else I didn't get you?Log in to reply

national standard examination of physicsmy bad i assumed you were, – Mvs Saketh · 2 years, 6 months agoLog in to reply

– Deepanshu Gupta · 2 years, 6 months ago

ohh That one! No I'am Not eligible for it . Since I passed 12th...!!Log in to reply