Worked examples
The question
Question: find the center, the vertices and the foci of x² over 25 + y² over 9 = 1.
x225 + y29 = 1
1 Find the center
There are no brackets: just x² and y².
So the center is (0, 0).
center = (0, 0)
2 Find a and b
The bigger bottom is a²: 25.
It is under x.
So the long side goes across.
bigger bottom = a²: under x → wide
a² = 25, so a = 5: 5 times 5 is 25.
a is half the long side.
a2 = 25 → a = 5
The smaller bottom is b² = 9, so b = 3.
b is half the short side.
b2 = 9 → b = 3
3 Vertices
The vertices are the ends of the long side: go a = 5 from the center, left and right.
vertices = (−5, 0) and (5, 0)
Draw the grid.
Put the center on the grid.
The two vertices, 5 each way.
And b = 3 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.
c2 = a2 − b2
Careful!
For an ellipse it is minus: c is shorter than a, so the foci stay inside.
Careful: not c² = 25 + 9 ✗
Now put the numbers in: a² = 25 and b² = 9.
c2 = 25 − 9
25 minus 9 is 16.
c2 = 16
4 × 4 = 16
So c = 4, because 4 times 4 is 16.
c = 4
Go c from the center along the long side.
The foci are (−4, 0) and (4, 0).
foci = (−4, 0) and (4, 0)
Put the foci on the grid.
5 Check
Let's check with the end of the short side, (0, 3).
Join it to both foci.
Each line goes 4 across and 3 up: Pythagoras.
c² plus b² is 16 plus 9: 25.
The square root is 5.
distance = √16 + 9 = √25 = 5
The two distances add up to 10, the length of the long side.
That is true for every point of an ellipse.
Right!
5 + 5 = 10, long side 2 × 5 = 10
Let's look back.
First: find the center.
Then: find a and b.
Then: vertices.
Then: find the foci.
Last: check.
Answer vertices = (−5, 0) and (5, 0)
▶ Watch it solved
The question
Question: find the center, the vertices and the foci of x² over 9 + y² over 25 = 1.
x29 + y225 = 1
1 Find the center
There are no brackets: just x² and y².
So the center is (0, 0).
center = (0, 0)
2 Find a and b
The bigger bottom is a²: 25.
It is under y.
So the long side goes up and down.
bigger bottom = a²: under y → tall
a² = 25, so a = 5: 5 times 5 is 25.
a is half the long side.
a2 = 25 → a = 5
The smaller bottom is b² = 9, so b = 3.
b is half the short side.
b2 = 9 → b = 3
3 Vertices
The vertices are the ends of the long side: go a = 5 from the center, down and up.
vertices = (0, −5) and (0, 5)
Draw the grid.
Put the center on the grid.
The two vertices, 5 each way.
And b = 3 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.
c2 = a2 − b2
Careful!
For an ellipse it is minus: c is shorter than a, so the foci stay inside.
Careful: not c² = 25 + 9 ✗
Now put the numbers in: a² = 25 and b² = 9.
c2 = 25 − 9
25 minus 9 is 16.
c2 = 16
4 × 4 = 16
So c = 4, because 4 times 4 is 16.
c = 4
Go c from the center along the long side.
The foci are (0, −4) and (0, 4).
foci = (0, −4) and (0, 4)
Put the foci on the grid.
5 Check
Let's check with the end of the short side, (3, 0).
Join it to both foci.
Each line goes 4 across and 3 up: Pythagoras.
c² plus b² is 16 plus 9: 25.
The square root is 5.
distance = √16 + 9 = √25 = 5
The two distances add up to 10, the length of the long side.
That is true for every point of an ellipse.
Right!
5 + 5 = 10, long side 2 × 5 = 10
Let's look back.
First: find the center.
Then: find a and b.
Then: vertices.
Then: find the foci.
Last: check.
Answer vertices = (0, −5) and (0, 5)
▶ Watch it solved
The question
Question: find the center, the vertices and the foci of (x − 1)² over 16 + (y + 2)² over 4 = 1.
(x − 1)216 + (y + 2)24 = 1
1 Find the center
The center comes from the brackets.
The sign turns: x − 1 gives 1, y + 2 gives −2.
So the center is (1, −2).
center = (1, −2)
x − 1 → h = 1
y + 2 → k = −2
2 Find a and b
The bigger bottom is a²: 16.
It is under x.
So the long side goes across.
bigger bottom = a²: under x → wide
a² = 16, so a = 4: 4 times 4 is 16.
a is half the long side.
a2 = 16 → a = 4
The smaller bottom is b² = 4, so b = 2.
b is half the short side.
b2 = 4 → b = 2
3 Vertices
The vertices are the ends of the long side: go a = 4 from the center, left and right.
vertices = (−3, −2) and (5, −2)
Draw the grid.
Put the center on the grid.
The two vertices, 4 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.
c2 = a2 − b2
Careful!
For an ellipse it is minus: c is shorter than a, so the foci stay inside.
Careful: not c² = 16 + 4 ✗
Now put the numbers in: a² = 16 and b² = 4.
c2 = 16 − 4
16 minus 4 is 12.
c2 = 12
So c is the square root of 12.
c = √12
But 12 has a square number in it: 4.
12 is 4 times 3.
c = √4 × 3
A root of a times splits into two roots.
c = √4 × √3
√4 is 2.
2 × 2 = 4
So c = 2√3.
That is the exact answer, made simple.
c = 2√3
Go c from the center along the long side.
The foci are (1 − 2√3, −2) and (1 + 2√3, −2).
foci = (1 − 2√3, −2) and (1 + 2√3, −2)
Put the foci on the grid.
5 Check
Let's check with the end of the short side, (1, 0).
Join it to both foci.
Each line goes 2√3 across and 2 up: Pythagoras.
c² plus b² is 12 plus 4: 16.
The square root is 4.
distance = √12 + 4 = √16 = 4
The two distances add up to 8, the length of the long side.
That is true for every point of an ellipse.
Right!
4 + 4 = 8, long side 2 × 4 = 8
Let's look back.
First: find the center.
Then: find a and b.
Then: vertices.
Then: find the foci.
Last: check.
Answer vertices = (−3, −2) and (5, −2)
▶ Watch it solved
