# Chapter 13 (Alg 2) Flashcards

1
Q

Distance formula

A

ⅆ=√((x_2−x_1 )^2+(y_2−y_1 )^2 )

2
Q

The midpoint Formula

A

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

3
Q

Focus

A

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

4
Q

Directrix

A

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

5
Q

Vertex

A

lies halfway between the focus and the directrix

6
Q

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

A

x^2 = 4py

7
Q

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

A

y^2 = 4px

8
Q

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

A

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)

9
Q

Circle

A

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

10
Q

A

distance between the center and any point on the circle

11
Q

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

A

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

12
Q

ellipse

A

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

13
Q

major axis

A

The line segment joining the vertices of an ellipse

14
Q

center of the ellipse

A

the midpoint of the major axis of an ellipse

15
Q

co-vertices of an ellipse

A

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

16
Q

minor axis

A

The line segment joining the co-vertices of an ellipse

17
Q

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

Horizontal

A

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

18
Q

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

Vertical

A

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

19
Q

hyperbola

A

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

20
Q

vertices (Hyperbola)

A

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

21
Q

Transverse axis

A

The segment joining the vertices of a hyperbola

22
Q

Midpoint (Hyperbola)

A

center of the hyperbola

23
Q

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

Horizontal

A
```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```
24
Q

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

Vertical

A
```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```
25
Q

Conic Sections or conics

A

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

26
Q
```Standard Equations of Translated Conics
Horizontal Axis (Circle is a circle regardless)```
A
```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```
27
Q
```Standard Equations of Translated Conics
Vertical Axis (Circle is a circle regardless)```
A

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

28
Q

General second-degree equation in x and y

A

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

29
Q

Classifying Conic Sections

A

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