Chapter 13 (Alg 2)

Question 1

Distance formula

Answer 1

ⅆ=√((x_2−x_1 )^2+(y_2−y_1 )^2 )

Question 2

The midpoint Formula

Answer 2

M=((x_1+x_2)/2,(y_1+y_2)/2)

Question 3

Focus

Answer 3

A point that lies on the axis of symmetry, lies p amount from vertex

Question 4

Directrix

Answer 4

A line Perpendicular to the axis of symmetry, lies p amount from vertex

Question 5

Vertex

Answer 5

lies halfway between the focus and the directrix

Question 6

Equation of parabola opening up or down with vetex (0,0)

Answer 6

x^2 = 4py

Question 7

Parabolas opening to the left or right with vertex (0,0)

Answer 7

y^2 = 4px

Question 8

Standard equation of a Parabola with Vertex at (0,0)

Answer 8

The standard form of the equation of a parabola with vertex at (0,0) is as follows: (the last one is axis of symm.)
x^2 = 4py, focus (0,p), Directrix y = -p, Vertical (x = 0)
y^2 = 4px, focus (p,0), Directrix x = -p, Horizon. (y = 0)

Question 9

Circle

Answer 9

set of all points P in a plane that are equidistant from a fixed point, called the center

Question 10

radius

Answer 10

distance between the center and any point on the circle

Question 11

Standard Equation of a Circle with Center at (0,0)

Answer 11

The standard form of the equation ofa circle with center at (0,0) and radius r is as follows:
x^2 + y^2 = r^2

Question 12

ellipse

Answer 12

the set of all points P in a plane such that the sum of the distances between P and two fixed points, called the foci, is a constant

Question 13

major axis

Answer 13

The line segment joining the vertices of an ellipse

Question 14

center of the ellipse

Answer 14

the midpoint of the major axis of an ellipse

Question 15

co-vertices of an ellipse

Answer 15

The points of intersection of an ellipse and the line perpendicular to the major axis at the center

Question 16

minor axis

Answer 16

The line segment joining the co-vertices of an ellipse

Question 17

The Standard Equation of an Ellipse with Center at (0,0)

Horizontal

Answer 17

x^2/a^2 + y^2/b^2 = 1
Major Axis is Horizontal
Vertices (±a,0)
Co-Vertices (0,±b)
The major and minor axes are lengths 2a and 2b, respectively, where a>b>0.
The foci of the ellipse lie on the major axis, c units from the center where c^2 = a^2 - b^2

Question 18

The Standard Equation of an Ellipse with Center at (0,0)

Vertical

Answer 18

x^2/b^2 + y^2/a^2 = 1
Major Axis is Vertical
Vertices (0,±a)
Co-Vertices (±b,0)
The major and minor axes are lengths 2a and 2b, respectively, where a>b>0.
The foci of the ellipse lie on the major axis, c units from the center where c^2 = a^2 - b^2

Question 19

hyperbola

Answer 19

the set of all points P such that the difference of the distances from P to the two foci is a constant

Question 20

vertices (Hyperbola)

Answer 20

the line through the foci intersects the hyperbola at the two vertices

Question 21

Transverse axis

Answer 21

The segment joining the vertices of a hyperbola

Question 22

Midpoint (Hyperbola)

Answer 22

center of the hyperbola

Question 23

Standard Equation of a Hyperbola with Center at (0,0)

Horizontal

Answer 23

x^2/a^2 −y^2/b^2 =1
Transverse Axis horizontal
Asymptotes y = ±b/a (x)
Vertices (±a,0)
The foci of the hyperbola lie on the transverse axis, c units from the center where c^2 = a^2 + b^2

Question 24

Standard Equation of a Hyperbola with Center at (0,0)

Vertical

Answer 24

y^2/a^2 −x^2/b^2 =1
Transverse Axis vertical
Asymptotes y = ±a/b (x)
Vertices (0,±a)
The foci of the hyperbola lie on the transverse axis, c units from the center where c^2 = a^2 + b^2

Question 25

Conic Sections or conics

Answer 25

Parabolas, circles, ellipses and hyperbolas are all curves that are formed by the intersection of a plane and a double cone

Question 26

Standard Equations of Translated Conics
Horizontal Axis (Circle is a circle regardless)

Answer 26

In the following equations the point (h,k) is the vertex of the parabola and the center of the other conics:
Circle: (x - h)^2 + (y - k)^2 = r^2
Parabola: (y - k)^2 = 4p(x - h)
Ellipse: (x−h^2/a^2 +(y−k)^2/b^2 =1
Hyperbola: (x−h^2/a^2 −(y−k)^2/b^2 =1

Question 27

Standard Equations of Translated Conics
Vertical Axis (Circle is a circle regardless)

Answer 27

In the following equations the point (h,k) is the vertex of the parabola and the center of the other conics:
Parabola: (x - h)^2 = 4p(y - k)
Ellipse: (x−h^2/b^2 +(y−k)^2/a^2 =1
Hyperbola: (y−k)^2/a^2 −(x−h^2/b^2 =1

Question 28

General second-degree equation in x and y

Answer 28

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
The equation of any conic can be written in this form
The Discriminant of the equation is B^2 - 4AC

Question 29

Classifying Conic Sections

Answer 29

If the graph of Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 is a conic, then the type of conic can be determined by the following characteristics.
Circle: B^2 - 4AC 0