Identify the coordinates of the center, the four vertices, and the two foci of this conic section.

Question

anonymous · Answer

I need help finding the 4 vertices and the 2 foci.

anonymous · Answer

Can you identify the major/horizontal axes and their lengths?

anonymous · Answer

major/minor axes*

anonymous · Answer

I don't really understand how to do that

anonymous · Answer

How did you determine this was an ellipse?

anonymous · Answer

Because it is using addition. if it's subtraction, then it is a hyperbole. That's just a trick that someone told me.

anonymous · Answer

Well, a circle uses addition too. How do you know it's not a circle?

anonymous · Answer

Standard form of a circle is (x-h)^2+(y-k)^2=r^2

anonymous · Answer

So what do you notice between the difference of the equation of a circle and the equation of an ellipse

anonymous · Answer

The equation of an ellipse is

anonymous · Answer

There isn't a radius

anonymous · Answer

In the equation of a circle centered at the origin, for example:

$$x^2 + y^2 = 2^2$$

What happens if you divide both sides by the radius squared?

anonymous · Answer

Then it looks like the equation of an ellipse

anonymous · Answer

So that tells you that a circle is just a special kind of ellipse. A circle is just an ellipse with the same major/minor axes. If you notice the numbers below the 'x^2' and the 'y^2' of the circle I just gave you, you see they are the same numbers. In an ellipse, you notice they are different. That's because they're the major/minor axes, and defines how long the ellipse is stretched out in each direction. If they are stretched out in both directions the same distance, it becomes a circle

anonymous · Answer

oh ok

anonymous · Answer

The largest number under the 'x' or 'y' is the major axis. In your case, the '16' under the 'y' is bigger than the '9' under the 'x' so you know the ellipse is shaped by being elongated along the y-axis and centered at the point you found was the center.

The numbers under the 'x' and 'y' are half the lengths of the respective axis squared.

So under the 'y' you have a 16, which is 4^2. So the length of the major axis is 8. The vertices of the major axis will be the coordinates at each of the endpoints of the major axis

anonymous · Answer

Okay that makes sense

anonymous · Answer

$$\frac{(x+2)^2}{3^2} + \frac{(y - 3)^2}{4^2} = 1$$

If you sketch it, it'll look something like:

|dw:1342368693169:dw|