Let O be the center and let F be one of the foci of the ellipse 25x^2 +16 y^2 = 400. A second ellipse, lying inside and tangent to the first ellipse, has its foci at O and F. What is the length of the minor axis of this second ellipse?

macar0ni Apr 3, 2021

#3**+2 **

The 'foci' equation of an ellipse is

\(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \)

a is the half the length of the horizontal axis and b is half the length of the vertical axis

(h,k) is the centre

If a is bigger than b then it will be more long, (the major axis will be horizontal)

If a is smaller than b then it will be more tall, (the major axis will be vertical)

So lets see what we have been given.

25x^2 +16 y^2 = 400

divide through by 400

\(\frac{x^2}{16}+\frac{y^2}{25}=1\\ \frac{x^2}{4^2}+\frac{y^2}{5^2}=1\\ \)

So the centre is (0,0)

It is a tall one.

The major axis will be from (-5,0) to (5,0)

The minor axis will be from (0,-4) to (0,4)

To find the focal points we use c where

\(c^2=|a^2-b^2|\\ c^2=|16-25|\\ c=3 \)

So the distance from the middle to the focal point is 3 units.

The foci will be at (0,-3) and (0,3)

We need to find the equation of the ellipse with

focal points (0,0) and (0,3) [I could have chosen (0,-3) if I had wanted to]

the centre will be (1,1.5)

c is the distance from the centre to the focal point, so c=1.5

It is just going to touch the other ellipse in one point and that point will be (0,5)

So the major axis b will be 5-1.5=3.5units

We have to find a

\(c^2=|a^2-b^2|\)

b >a so

\(c^2=b^2-a^2\\ a^2=b^2-c^2\\ a^2=3.5^2-1.5^2\\ a^2=10\\ a=\sqrt{10} \)

So the major axis is 2*3.5=7 units long

and

the minor axis will be **2sqrt10** which is approx 6.32 units long

For anyone interested: the equation of the second ellipse will be

\(\frac{x^2}{10}+\frac{(y-1.5)^2}{12.25}=1\)

Melody Apr 4, 2021

#4**+1 **

Thank you for the explanation, I think I understand the question. :DDD

I'm sorry if this is a dumb question, but what's the importance/point of a foci point?

=^._.^=

catmg
Apr 4, 2021

#5**+2 **

It is definitely not a dumb question.

It is very important to understand the relevance of a focal point.

__A circle__ is the set of all points equidistant from the central point.

__An ellipse__ is the set of all points where the sum of the distance to each of the focal points stays constant.

So

A circle is an ellipse where the 2 foci are in the same spot.

Think of the way this guy draws the ellipse in the video below and you will see what I mean.

Here is how to draw one with two pins and a piece of string

Here is a great site that covers many important characteristics of ellipses.

There is a number of interactive pictures. Play with them make sure you understand what they are trying to demonstrate to you.

Melody
Apr 5, 2021

#6**+1 **

Thank you for responding. :DDD

Ohhh I get it, it's like the average of the foci points.

I shall start studying ellipses on alcumus.

I think we did a similar activity during my science class when we were learning about planets.

Sadly, it failed quite misreably since it was hard to get everything organized while in quaratine.

I miss doing labs. :((

Last year, we were supposed to do this forensic lab where we would try to solve a m****r case.

Our teacher had to make everything electronic and it was still fun, but we didn't get to do things such as collect finger prints.

=^._.^=

catmg
Apr 5, 2021