10TH GRADE · EM2 · CHAPTER 10

Conics

Equations, Graphing & Identifying

12 slides Conics Test Prep

Conic Sections

Slicing a cone:

Shape	Equation	Key
Circle	(x−h)²+(y−k)²=r²	r=r
Ellipse	(x−h)²/a²+(y−k)²/b²=1	a≠b
Hyperbola	(x−h)²/a²−(y−k)²/b²=1	−
Parabola	y=a(x−h)²+k	1 sq

Circle Equation

(x−h)² + (y−k)² = r²

Center (h,k)

Radius r

(x−3)²+(y+2)²=25

C(3,−2), r=5

Complete the Square

x²+y²−6x+4y−12=0

Group x's and y's

Complete squares

(x−3)²+(y+2)²=25

C(3,−2), r=5

Ellipse

(x−h)²/a² + (y−k)²/b² = 1

Center (h,k)

a = horiz, b = vert

a>b horiz, b>a vert

a=4, b=3

Hyperbola

Horizontal: x²/a² − y²/b² = 1

Vertical: y²/a² − x²/b² = 1

Positive term = direction

Asymptotes: ±b/a slope

Identify Conics

General form:

Condition	Conic
A = B	Circle
A≠B, same sign	Ellipse
Opposite signs	Hyperbola
One sq term	Parabola

Parabola as Conic

Vertical: (x−h)²=4p(y−k)

Horizontal: (y−k)²=4p(x−h)

p = vertex to focus

Focus (h,k+p)

Directrix y=k−p

p=2, Focus(1,4)

Graphing Tips

1. Center

2. Type

3. a and b

4. Plot points

5. Asymptotes first

Practice Problems

1. Circle: center & radius?

▶ Show Answer

C(−2,5), r=3

2. What conic?

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Ellipse

3. What conic?

▶ Show Answer

Hyperbola

4. Parabola details?

▶ Show Answer

V(−1,3), F(−1,6), y=0

Common Mistakes

Sign in (x−h)

Both x AND y

+ ellipse, − hyperbola

r² vs r

Chapter 10 Summary

✔ Circle

✔ Ellipse

✔ Hyperbola

✔ Parabola (conic)

✔ Identify conics

Good luck!