Step by step
The question
Question: find the center, the vertices and the foci of (x + 1)² over 4 + (y + 3)² over 9 = 1.
1 Find the center
The center comes from the brackets.
The sign turns: x + 1 gives −1, y + 3 gives −3.
So the center is (−1, −3).
2 Find a and b
The bigger bottom is a²: 9.
It is under y.
So the long side goes up and down.
a² = 9, so a = 3: 3 times 3 is 9.
a is half the long side.
The smaller bottom is b² = 4, so b = 2.
b is half the short side.
3 Vertices
The vertices are the ends of the long side: go a = 3 from the center, down and up.
Draw the grid.
Put the center on the grid.
The two vertices, 3 each way.
And b = 2 the other way: the ends of the short side.
A smooth oval through the four points.
4 Find the foci
The two foci sit inside, on the long side.
Their distance c from the center: c² = a² − b².
Careful!
For an ellipse it is minus: c is shorter than a, so the foci stay inside.
Now put the numbers in: a² = 9 and b² = 4.
9 minus 4 is 5.
So c is the square root of 5.
5 is not a square number.
So we keep √5.
That is the exact answer.
It is about 2.2.
Go c from the center along the long side.
The foci are (−1, −3 − √5) and (−1, −3 + √5).
Put the foci on the grid.
5 Check
Let's check with the end of the short side, (1, −3).
Join it to both foci.
Each line goes √5 across and 2 up: Pythagoras.
c² plus b² is 5 plus 4: 9.
The square root is 3.
The two distances add up to 6, the length of the long side.
That is true for every point of an ellipse.
Right!
Let's look back.
First: find the center.
Then: find a and b.
Then: vertices.
Then: find the foci.
Last: check.