Identify a conic from its equation

Warm-up

Three quick questions from earlier lessons.

Discover

$4x^2 + 4y^2 = 36$ . Which conic is it?

A circle
A hyperbola

Key idea

The squared terms tell which conic an equation draws.

Only one variable squared: parabola. Both squared, same sign: ellipse (a circle if the coefficients are equal). Both squared, opposite signs: hyperbola.

Worked example

Identify $9y^2 - x^2 = 9$ , then put it in standard form.

A hyperbola

Both squared, opposite signs.

y^2 - \dfrac{x^2}{9} = 1

Divide by 9.

Branches open up and down, vertices

(0, \pm 1)

y² is positive.

Your turn, guided

Your turn: $x^2 + 4y^2 = 16$ .

An ellipse

Both squared, same sign, different coefficients.

a = 4

: vertices

(\pm 4, 0)

Divide by 16: x²/16 + y²/4 = 1. Type a.

Practice ★

What kind of conic is $2x^2 + 3y^2 = 36$ ?
1. An ellipse
2. A circle
3. A hyperbola
4. A parabola
What kind of conic is $2x^2 + 7y^2 = 67$ ?
1. An ellipse
2. A circle
3. A hyperbola
4. A parabola

Watch out!

Practice ★★

What kind of conic is $9x^2 - 5y^2 = 13$ ?
1. A hyperbola
2. An ellipse
3. A parabola
What kind of conic is $4x^2 - 2y^2 = 40$ ?
1. A hyperbola
2. An ellipse
3. A parabola

What kind of conic is $y = 5x^2 - 9x$ ?
1. A parabola
2. A hyperbola
3. A circle
What kind of conic is $y = 9x^2 - 4x$ ?
1. A parabola
2. A hyperbola
3. A circle

Exit ticket

Three questions to check you have got it.

What kind of conic is $2x^2 + 3y^2 = 36$ ?
1. An ellipse
2. A circle
3. A hyperbola
4. A parabola
What kind of conic is $2x^2 + 7y^2 = 67$ ?
1. An ellipse
2. A circle
3. A hyperbola
4. A parabola
What kind of conic is $6x^2 + 5y^2 = 78$ ?
1. An ellipse
2. A circle
3. A hyperbola
4. A parabola

Summary card

Which conic?

One variable squared: parabola.
Both, same sign: ellipse (equal coefficients: circle).
Both, opposite signs: hyperbola.

Watch out: look at the squared terms, not the constant.

About this lesson