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 a2 .Then it’s a Vertical Ellipse(look at the formula of the Vertical Ellipse ).
b2 = 9 → b = 3 ; a2 = 16 → a = 4
Remember b2 + c2 = a2 so c2 = 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 F1, take the coordinate of the centre subtract c value to find F2.But remember this is the Vertical Ellipse. For example if we have a center C(h,k) and we found c so F1=(h,k+c) and F2=(h,k-c)
But if it’s Horizontal Ellipse, we will have F1=(h+c,k),F2=(h-c,k) .I’ll show you an example later.....
Come back with the first question , we’ll have the point F1(2,-1+√7) and F2(2,-1-√7).Now let’s find the vertices . To find A1 and A2 in the vertical ellipse,A1=(h,k+a) and A2=(h,k-a). And B1=(h+b,k);B2=(h-b,k)
Then A1= (2,3),A2= (2,-5)
B1=(5,-1) ,B2= (-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 → a2 = 25, b2 = 16 and it’s a Horizontal Ellipse.
→ a = 5 , b = 4
c2 = a2- b2 = 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 :
F1=(h+c,k),F2=(h-c,k)
A1= (h+a,k),A2= (h-a,k)
→F1=(0,2),F2=(-6,2)
A1=(2,2),A2=(-8,2)
B1=(-3,6),B2=(-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 :
16x2 + 25y2 + 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 x2 and y2 in this formula and in the parabola, there is x 2 or y2 (just one of them).Second, there are same numbers in x2 and y2 with the circle(Ex: 16x2 and 16y2) 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 !