Graphing Conic Sections Part 2: Ellipses

The next conic section is like a circle, but it's like someone stretched a circle out in a particular direction. It's the ellipse! These are very important in astronomy as most celestial objects follow elliptical paths. Let's learn all about ellipses and how to graph them.

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[suspendedAnimations]

Every time I see this shape I think of the PHP 5 logo.

[srishti_chaurasia_]

Almost one third of the course to this whole Mathematics series, want to confess... had I understood all this in this way 15 years back when it was taught to me in my school, I would have been a different person today. Alas! teachers took no interest in making average students comfortable in Maths classes. They were happy with so called brighter ones in class.

Undoing the harm done that time by understanding all this now from you. Lucky that stumbled upon your account. Kudos to you @ProfessorDaveExplains. 🤗

[autumnvanhoose]

you're my lifesaver this week between chem and pre calc. me and my roomie <3 you

[Firnas_]

brothers in the comprehension part 2nd question if anyone have problem just draw graph
it is actually
c= distance from center to focus(foci), it is 1 from both x and y axis when you draw graph you can see ;
a=distance from center to vertex which is 5
remember center is not (0, 0) rather (-2, 4)
so count everything from that position.

in the last part, since y axis tall, it is taller ellipse, that is why the formula will be, (x-h)²/b² + (y-k)²/a² = 1

hope this helps!

[paulahernandez]

When we are looking at the foci and the vertices to be plugged into c^2= a^2 + b^2, is there a particular order in which the points need to be plugged in? For example, does the x value of the first or second parenthesis of the foci go in for a^2 or not? The same question goes for the vertices with c^2. I would appreciate the help. Thanks

[harshsinghal]

how do we derive the standard form of the equation of ellipse?

[BebeIsGoated]

The standard form equation for an ellipse centered at $(h, k)$ is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, where $a$ is the distance from the center to a vertex along the major axis and $b$ is the distance from the center to a co-vertex along the minor axis.

In this case, the center is given as $(-2, 4)$, so $h=-2$ and $k=4$. Additionally, we know the foci and vertices, which provide key information about the dimensions of the ellipse:

- The distance between the center and each focus is $c$, so $c = 5 - 4 = 1$.
- The distance between the center and each vertex along the major axis is $a$, so $a = 9 - 4 = 5$.
- The distance between the center and each co-vertex along the minor axis is $b$, so $b = 1$.

Now we have all the information we need to write the equation:

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

This is the equation in standard form of an ellipse with center (-2, 4), foci at (-2, 3) and (-2, 5), and vertices at (-2, -1) and (-2, 9)

[jenniferrubyj]

Sir can u please explain how c square =1 in the comprehension 2nd problem

[BoZhaoengineering]

If you could add how to derive the equation of ellipse, this video would be perfect. And all function transformation could apply for ellipse if we only take one fourth of ellipse as function. Thus, all ellipse is applicable for transformation.

[cat-uqhw]

2:51 when do we use c^2 = a^2 - b^2 and c^2 = a^2 + b^2? You should've put that before introducing this equation.

[nahaaryounus]

Thank you so much for uploading this!!

[ignacioleikis]

Can someone explain how to solve the second one?

[manoaalix]

Excellent video ! Thanks professor Dave !

[papillonnage]

I think I have found a tiny mistake.
The second equation (in checking comprehension) i think 25 and 24 have been swapped in the final result. @5:53
Awesome channel, btw.

[revil]

Thanks Prof Dave for explaining the topic very clearly! You have been very helpful! Keep up the good work!

[arway]

youtube please put his conic section videos up first when I search the topic, not the one hour long videos!
I had to search for your channel specifically I feel bad for all the students who haven't found you yet : (

[Teiwaz]

Anyone else watching and seeing how the 2nd comprehension example is almost a proof for why every sphere is an ellipse? 😁

To clarify imagine said equation being
(y - 4)²/25 + (x + 2)²/25 = 1

Multiply by 25
(y - 4)² + (x + 2)² = 25

And you got the standard form of a sphere 😉
(x - h)² + (y - k)² = r²
Or based on the equation from before
(x + 2)² + (y - 4)² = 25

So remember: if your ellipse equation has a = b it's actually a sphere, not only an ellipse.
Also pretty obvious if you look at it as a graph ( 1:40 ) ofc.

[MichaelLeightonsKarlyPilkboys]

You are a hero Professor Dave. Thanks so much for these.

[stephenhemingway]

Will you be explaining eccentricity in another video beyond telling us what the values are for various types of conic sections?

[lianderferraraccio]

I had not understood this tutorial so i just skipped so i could come back later, just came back right after watching the hyperbola one! (the hyperbola and the ellipse concepts are very similar)