# Elipse in a semi circle

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

Note by Mvs Saketh
5 years, 11 months ago

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My $e$ 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.

- 5 years, 11 months ago

- 5 years, 11 months ago

Here's the solution.

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))

$\Rightarrow e=\sqrt{\frac{2}{3}}$

- 5 years, 11 months ago

Great !!! Thanks a lot ! @Ronak Agarwal But Ronak shoudn't it is b(sin(t)) instead of b(1+ sin(t) ) Plz Explain me ! Thanks

- 5 years, 11 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.

- 5 years, 11 months ago

Thanks ! Now I got it completely ! I really appreciate your solution very much !!

- 5 years, 11 months ago

Thanks

- 5 years, 11 months ago

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

- 5 years, 11 months ago

- 5 years, 11 months ago

- 5 years, 11 months ago

How much marks are you getting @Mvs Saketh . I'm just asking.

- 5 years, 11 months ago

Really bad, should have done level 2 chem instead of math,,, infact its shamefully 58, It was an assymetric distribution with physics way too easy and maths part 2 hard for me,

- 5 years, 11 months ago

I am already a KVPY scholar hence I haven't given KVPY this year.

- 5 years, 11 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?

- 5 years, 11 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.

- 5 years, 11 months ago

Well IISC is perhaps the best place for BSc MSc Phd in india,, ofcourse CMI, DAE are no less, but IISC has international recognition as far as i know,, and i think oppurtunities depend on how much u are willing to put in,,

And if u want to spend the rest of your life wondering,, i think its awesome ,,,

- 5 years, 11 months ago

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

- 5 years, 11 months ago

I think u will excel in whatever u do!

- 5 years, 11 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)

- 5 years, 11 months ago

Hey! which question are you talking about??

- 5 years, 11 months ago

Maths Part-2, you can see this question there.

- 5 years, 11 months ago