Ch 9.1 Circles and Parabolas
Date:_______________
Essential Question: How can you recognize conic sections and solve problems using circles and parabolas?
Topic/Question
Notes
An Overview of
A Conic Section(Conic) is
Conic Sections
The 4 Basic Non-Degenerate Conics
Degenerate Conics
The two way to
define Conics:
By a General Equation:
By a Locus of Points:
Warming Up to
Circles
1) What is the distance formula and where does it come from?
2) Find the distance between (1, 1) and (4, 5)
3) Find the distance r between (h, k) and (x, y) in the figure.
Definition of a
Circle
The set of all points (x ,y) in a plane that are _________ from a fixed point (h, k)
called the center of the circle.
The distance between any point (x, y) and the center is called the ________
EX 1: Sketching a
Circle
(x - __)² + (y - __)² = r ² where (___, ___) is the center and ____ is the radius
Sketch 𝑥 2 + 𝑦 2 = 16 and (𝑥 − 4)2 + (𝑦 − 2)2 = 16
EX 2: Writing an
Equation of a
Circle
You Try!
EX 3: Graphing a
Circle
The Point (1, 4) is on a circle whose center is at (-2, 3). Write the
standard form of the equation and graph.
The Point ( 1, -2) is on a circle whose center is at (-3, -5).
Write the standard form of the equation and graph
Sketch the circle given by the equation
𝑥 2 − 6𝑥 + 𝑦 2 − 2𝑦 + 6 = 0
You Try!
EX 4:
You Try!
Sketch the circle given by the equation
𝑥 2 − 10𝑥 + 𝑦 2 − 6𝑦 + 25 = 0
Put the circle given by the equation 9𝑥 2 + 9𝑦 2 + 54𝑥 − 36𝑦 + 17 = 0 in standard form
Put the circle given by the equation 4𝑥 2 + 4𝑦 2 + 12𝑥 − 24𝑦 + 41 = 0 in standard form
Parabolas
A parabola is the set of all points (x ,y) in a plane that are
equidistant from a fixed line called the directrix and a fixed
point called the focus
The midpoint between the directrix and focus is the ______
1
Vertical Parabola: 4𝑝 (x – h)² = (y-k) where p≠0
1
Horizontal Parabola: 4𝑝 (y – k) ² = (x - h) where p≠0
1
Let 4𝑝 = a
EX 1: Find the
standard form of
the parabola
You Try!
EX 2: Find the
standard form of
the parabola
You Try!
Parabolas in the form of y = a(x-h)² + k ________if a > 0 and ________ if a < 0
Parabolas in the form of x = a(y-h)² + k ________ if a > 0 and ________ if a < 0
Find the standard form of the parabola with vertex at the origin and focus at (0,4)
Find the standard form of the parabola with vertex at the origin and focus at (-2,0)
Find the standard form of the equation of a parabola with vertex at (1, 0) and focus at (2, 0)
Find the standard form of the equation of a parabola with a vertex at (-1, 2) and focus at
(-1, 0)
EX 3: Finding a
focus
1
1
Find the focus of the parabola given by y = − 2x² - x + 2
Put application WS
here.