Step by step
The question
Question: find the center, the vertices and the foci of x² + 9y² = 9.
1 Make the right side 1
The standard form has 1 on the right.
So we divide every part by 9.
We do it on both sides, so they stay equal.
9/9 is 1/1.
So x² over 9 + y² = 1.
2 Find the center
There are no brackets: just x² and y².
So the center is (0, 0).
3 Find a and b
The bigger bottom is a²: 9.
It is under x.
So the long side goes across.
a² = 9, so a = 3: 3 times 3 is 9.
a is half the long side.
The smaller bottom is b² = 1, so b = 1.
b is half the short side.
4 Vertices
The vertices are the ends of the long side: go a = 3 from the center, left and right.
Draw the grid.
Put the center on the grid.
The two vertices, 3 each way.
And b = 1 the other way: the ends of the short side.
A smooth oval through the four points.
5 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 = 1.
9 minus 1 is 8.
So c is the square root of 8.
But 8 has a square number in it: 4.
8 is 4 times 2.
A root of a times splits into two roots.
√4 is 2.
So c = 2√2.
That is the exact answer, made simple.
Go c from the center along the long side.
The foci are (−2√2, 0) and (2√2, 0).
Put the foci on the grid.
6 Check
Let's check with the end of the short side, (0, 1).
Join it to both foci.
Each line goes 2√2 across and 1 up: Pythagoras.
c² plus b² is 8 plus 1: 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: make the right side 1.
Then: find the center.
Then: find a and b.
Then: vertices.
Then: find the foci.
Last: check.