Chapter 10 Conics
10.2 Circles
Standard Form of a Circle
General Form
Graph
Ex 1: Write the standard form of the equation of the circle that has
a. Center at (2,4) and passes through (−8, 15)
b. Center at (1, −2)and is tangent to the line 𝑦 = 6
Ex 2: Rewrite 𝑥 2 + 𝑦 2 − 4𝑥 + 14𝑦 − 47 = 0 into standard form.
Ex 3: Write the equation of the circle in standard form that passes through (0, −1), (−3, −2), (−6, −1)
10.3 Ellipse
Standard Form of Ellipse
Ex 1: Graph
(𝑥−1)2
16
+
(𝑦+2)2
36
=1
Ex 2: Graph
foci.
(𝑥−1)2
25
+
(𝑦−1)2
4
= 1 and state the coordinates of the
Eccentricity of an Ellipse
Ex 3: Write 16𝑥 2 + 25𝑦 2 − 96𝑥 − 200𝑦 + 144 = 0 in standard form.
Ex 4: Write the equation of an ellipse in standard form given it has
a. Center (2,5), vertical ellipse, semi minor axis 4 units, major axis is 20 units
b. Center at (−4,2), foci at (−4, 2 ± √13), minor vertex at (−5,2) and (−3,2)
1
c. Major vertices at (7,15) and (7,3), eccentricity = 4
10.4 Hyperbola
Standard Form of Hyperbolas
Ex 1: Graph
(𝑦−2)2
4
+
(𝑥−1)2
16
=1
Ex 2: Write −4𝑥 2 + 9𝑦 2 − 24𝑥 − 90𝑦 + 153 = 0 in standard form.
Ex 3: Write the equation of a hyperbola in standard form given that it has
a. Vertices at (2, −1) and (−8, −1), conjugate axis is 4 units long
b. Foci (2, −3) and (2,7), vertices (2, −1) and (2,5)
10.6 Parabolas
Standard form of Parabolas
Ex 1: Graph 3𝑥 2 − 30𝑦 − 18𝑥 + 87 = 0. Identify the
coordinates of the vertex, focus, the equation of the directrix and
the axis of symmetry.
Ex 2: Graph (𝑦−1)2 = 8(𝑥 + 3). Identify vertex, focus, directrix
and axis of symmetry.
Ex 3: Write the equation of the parabola in standard form given that it’s focus is at (4, −2) and the equation
of the directrix is 𝑦 = 6
10.6 Rectangular and Parametric Forms of Conic Sections
Identifying Conic Sections from general form
Ex 1: Identify the conic section represented by each equation
a. 7𝑥 2 + 2𝑦 2 − 4𝑥 + 51𝑦 − 65 = 0
b. −10𝑥 2 + 8𝑦 2 −
25
3
1
𝑥 + 2 𝑦 − √7 = 0
c. −5𝑥 2 − 5𝑦 2 − 16𝑥 + 4 = 0
d. 𝑥 2 + 𝑦 − 3𝑥 + 4 = 0