The question
Question: name the conic and write in standard form 4x² + 9y² − 36 = 0.
4x2 + 9y2 − 36 = 0
1 Name the conic
Look at the squares.
x² and y² have the same sign, but 4 and 9 are different numbers.
So it is an ellipse.
only x² or only y² → parabola
x² and y², same number in front → circle
same signs, different numbers → ellipse ✓
different signs → hyperbola
2 Move the number
Move the plain number to the right: − 36 becomes + 36.
4x2 + 9y2 = 36
3 Make the right side 1
Divide every part by 36, so the right side is 1.
4/36 is 1/9.
9/36 is 1/4.
x29 + y24 = 1
36 ÷ 4 = 9, 36 ÷ 9 = 4
That is the standard form of an ellipse with center (0, 0).
ellipse: center (0, 0)
4 Check
Let's check with a point on it: (3, 0), a vertex.
Put it in the first equation.
4 × 3² + 9 × 0² − 36 = 36 + 0 − 36 = 0
We get 0, like the first equation says.
So the standard form is right!
Let's look back.
First: name the conic.
Then: move the number.
Then: make the right side 1.
Last: check.
Answer 36 ÷ 4 = 9, 36 ÷ 9 = 4
▶ Watch it solved
The question
Question: name the conic and write in standard form x² − 6x − 8y + 1 = 0.
x2 − 6x − 8y + 1 = 0
1 Name the conic
Look at the squares.
Only x² is here.
So it is a parabola.
only x² or only y² → parabola ✓
x² and y², same number in front → circle
same signs, different numbers → ellipse
different signs → hyperbola
2 Keep x on the left
Keep the x terms on the left.
Move − 8y and the plain number to the right: their signs change.
(x2 − 6x) = 8y − 1
3 Complete the square for x
Half of −6 is −3.
(−3) times (−3) is 9.
Add 9 to both sides.
(x2 − 6x + 9) = 8y − 1 + 9
−6 ÷ 2 = −3, (−3) × (−3) = 9
Careful!
What we add on the left, we add on the right too.
Then the two sides stay equal.
Careful: add 9 on both sides
4 Write as a square
The bracket is a full square: (x − 3)².
On the right, −1 plus 9 is 8.
(x − 3)2 = 8y + 8
5 Take out the number in front
Take 8 out of the right side: 8(y + 1).
That is the standard form of a parabola.
(x − 3)2 = 8(y + 1)
8 × (y + 1) = 8y + 8
6 Check
Let's check with the vertex, (3, −1).
Put it in the first equation.
3² − 6 × 3 − 8 × (−1) + 1 = 9 − 18 + 8 + 1
= 0
We get 0, like the first equation says.
So the standard form is right!
Let's look back.
First: name the conic.
Then: keep x on the left.
Then: complete the square for x.
Then: write as a square.
Then: take out the number in front.
Last: check.
Answer 8 × (y + 1) = 8y + 8
▶ Watch it solved
The question
Question: name the conic and write in standard form 4x² + 9y² − 16x + 18y − 11 = 0.
4x2 + 9y2 − 16x + 18y − 11 = 0
1 Name the conic
Look at the squares.
x² and y² have the same sign, but 4 and 9 are different numbers.
So it is an ellipse.
only x² or only y² → parabola
x² and y², same number in front → circle
same signs, different numbers → ellipse ✓
different signs → hyperbola
2 Group x and y
Put the x terms together in a bracket, and the y terms together.
(4x2 − 16x) + (9y2 + 18y) − 11 = 0
Move the plain number to the right: − 11 becomes + 11.
(4x2 − 16x) + (9y2 + 18y) = 11
3 Take out the number in front
Inside each bracket, x² and y² must have 1 in front.
So we take out 4 and 9.
4(x2 − 4x) + 9(y2 + 2y) = 11
4x² − 16x = 4(x² − 4x)
9y² + 18y = 9(y² + 2y)
4 Complete the square for x
Half of −4 is −2.
(−2) times (−2) is 4.
We add 4 inside the bracket.
But the bracket is multiplied by 4.
So on the left we really added 4 times 4: 16.
We add 16 on the right too.
4(x2 − 4x + 4) + 9(y2 + 2y) = 11 + 16
−4 ÷ 2 = −2, (−2) × (−2) = 4, 4 × 4 = 16
Careful!
The 4 is inside a bracket with 4 in front.
It counts 4 times: add 16.
Careful: not add 4 on the right ✗
5 Complete the square for y
Half of 2 is 1.
1 times 1 is 1.
We add 1 inside the bracket.
But the bracket is multiplied by 9.
So on the left we really added 9 times 1: 9.
We add 9 on the right too.
4(x2 − 4x + 4) + 9(y2 + 2y + 1) = 11 + 16 + 9
2 ÷ 2 = 1, 1 × 1 = 1, 9 × 1 = 9
6 Write as squares
Each bracket is now a full square.
On the right, 11 + 16 + 9 is 36.
4(x − 2)2 + 9(y + 1)2 = 36
7 Make the right side 1
Divide every part by 36, so the right side is 1.
4/36 is 1/9.
9/36 is 1/4.
(x − 2)29 + (y + 1)24 = 1
36 ÷ 4 = 9, 36 ÷ 9 = 4
That is the standard form of an ellipse with center (2, −1).
ellipse: center (2, −1)
8 Check
Let's check with a point on it: (5, −1), a vertex.
Put it in the first equation.
4 × 5² + 9 × (−1)² − 16 × 5 + 18 × (−1) − 11
= 100 + 9 − 80 − 18 − 11 = 0
We get 0, like the first equation says.
So the standard form is right!
Let's look back.
First: name the conic.
Then: group x and y.
Then: take out the number in front.
Then: complete the square for x.
Then: complete the square for y.
Then: write as squares.
Then: make the right side 1.
Last: check.
Answer 36 ÷ 4 = 9, 36 ÷ 9 = 4
▶ Watch it solved
