First thing I want to tell you is there are Horizontal Ellipse and Vertical Ellipse .The Formula of the two Ellipses are different.

First is the center of ellipse. We have(x-2)^{2} and (y+1)^{ 2} →the coordinate of the center is (2,-1) or we can call the point C_{(2,-1)}

Now, look at 9 and 16 . 9 <>2 and 16 is a^{2} .Then it’s a Vertical Ellipse(look at the formula of the Vertical Ellipse ).^{}

b^{2 }= 9 → b^{ }= 3 ; a^{2 }= 16 → a = 4

Remember b^{2 }+ c^{2} = a^{2} so c^{2} = 16 – 9 = 7 → c = √7

Ok so now we have to find the foci . the way to find the foci is easy. Just take the coordinate of the center plus c value to find F_{1}, take the coordinate of the centre subtract c value to find F_{2}.But remember this is the Vertical Ellipse. For example if we have a center C(h,k) and we found c so F_{1}=(h,k+c) and F_{2}=(h,k-c)

But if it’s Horizontal Ellipse, we will have F_{1}=(h+c,k),F_{2}=(h-c,k) .I’ll show you an example later.....

Come back with the first question , we’ll have the point F_{1}(2,-1+√7) and F_{2}(2,-1-√7).Now let’s find the vertices . To find A_{1} and A_{2} in the vertical ellipse,A_{1}=(h,k+a) and A_{2}=(h,k-a). And B_{1}=(h+b,k);B_{2}=(h-b,k)

Then A_{1}= (2,3),A_{2}= (2,-5)

B_{1}=(5,-1) ,B_{2}= (-1,-1)

Major Axis = 2a = 8;Minor Axis = 2b = 6

Then we’ll have a “beautiful” ellipse like the picture below (I did’nt draw it) !

This is the Focal Radii Property:

Another question:

Sketch the Ellipse and find the coordinates of the center, foci and vertices with this one:

First is find the center just like the first question . We have C(-3,2)

Second, take a look at 25 and 16 . 25 > 16 → a^{2 }= 25, b^{2} = 16 and it’s a Horizontal Ellipse.

→ a = 5 , b = 4

c^{2 }= a^{2}- b^{2} = 25-16 = 9 → c = 3

As I told you before, I’ll show you an example of the Horizontal Ellipse. And here is the formula :

F_{1}=(h+c,k),F_{2}=(h-c,k)

A_{1}= (h+a,k),A_{2}= (h-a,k)

→F_{1}=(0,2),F_{2}=(-6,2)

A_{1}=(2,2),A_{2}=(-8,2)

B_{1}=(-3,6),B_{2}=(-3,-2)

Major axis = 2a = 10

Minor axis = 2b = 8

Standard form → General form → Back again :

The last thing I want to tell you is how to recognize the parabola, circle and ellipse :

If they give you one formula like this :

16x^{2 }+ 25y^{2 }+ 96x - 100y - 156 = 0

So this one is the formula of the parabola , circle or ellipse.

First, it’s cannot be the parabola because there are x^{2} and y^{2 }in this formula and in the parabola, there is x ^{2} or y^{2 }(just one of them).Second, there are same numbers in x^{2} and y^{2} with the circle(Ex: 16x^{2} and 16y^{2}) but in this formula there are different numbers (16 and 25).So this one is the formula of an ellipse!

Ok so my mission is completed . Just remember to bring your paper with the circle and 30-35 points to our class tomorrow !