+132 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?
+118673 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
\(\frac{x^2}{10}+\frac{(y-1.5)^2}{12.25}=1\)
+2407 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?
=^._.^=
+118673 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.
+2407 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.
=^._.^=
