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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?

Apr 3, 2021

#3
+113190
+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

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

Apr 4, 2021
#4
+1337
+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
+113190
+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

https://youtu.be/Et3OdzEGX_w

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.

https://www.mathsisfun.com/geometry/ellipse.html

Melody  Apr 5, 2021
#6
+1337
+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
#7
+113190
+2

The centre is half way between the focal points, if that is what you mean.

Melody  Apr 6, 2021