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(MATH) > Analytical Geometry > Flashcards

Analytical Geometry Flashcards

1
Q

x coordinate is also called as

A

abscissa (reminds you of NANA OR abscess)

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2
Q

Y COORDINATE IN CARTESIAN SYSTEM IS AKA

A

ORDINATE

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3
Q

HOW TO SOLVE FOR THE SLOPE

A 
m= tanθ
m= y2-y1 / x2-x1
m= -A/B        ; A as coeff of X, while B of Y
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4
Q

COMPRATE THE SLOP OF LINES THAT ARE PARALLEL AND IN PERPENDICULAR

A

PARALLEL
m1=m2

PEERPENDICULAR
m1m2= -1

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5
Q

6 ways to express a line equation

A

GENERAL EQUATION
* Ax +By + C = 0

SLOPT INTERCEPT FORM
*(y-y1)=m(x-x1)

PONT SLOPE FORM
* y= mx + b , where b is the y intercept (0,b)

INTERCEPT FORM (2 intecepts)
* x/a + y/b = 1

two point form, *if given only two points ,

NORMAL FORM
*xcos θ + y sin θ = P
where P is the length of the line perpendicular to the point of origin and θ is the inclination of P.

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6
Q

FORMULAE FOR THE DISTANCE BETW TWO POINTS
FOR XY PLANE
& XYZ PLANE.

A

FOR XY-PLANE
* use phytagorean theorem.

FOR XYZ PLANE

  • use phytagorean theorem
  • x^2 + y^2 + z^2 = d^2
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7
Q

FORMULAE FOR THE DISTANCE BETW A LINE AND A POINT

A

given should be in general eq. Ax + By + C = 0
and point in (x1 , y1)

d= (Ax1 + By1 + C) / ±√(A^2 + B^2)

the symbol ± will be positive if
(imaging the line moving clock wise, if the line hits the point then the symbol will be +)

Analytical Geometry 09-02

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8
Q

perpendiculat distance betwee two parallel line!!!

A

Ax + By + C1 = 0 — line 1
Ax + By + C2= 0 —– line 2

coefficient of line 1 and 2 must be the same, then
d = c1 - c2 / ±√(A^2 + B^2)

± same as numerator symbol

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9
Q

formula for the angle betwee to lines!

A

given, m= tanθ , θ =Tan^-1(m)
solve for θ2 & θ1

θ = | θ2- θ1 |

the 2 angle between is;
θ & 180-θ

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10
Q

in the division of line segments, what are the two tyoes of line division and explain

A

internal and external division.

where the point (x,y) is within the starting and ending point then it is internal,

when the point (x,y) is an extension point of the start to end point then it is external

that ratio is is important in this

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11
Q

different ways to express the ratio of the point (x,y) in a line segment with respect to the start and end point of the line.

A

m: n
- where m is the distance betw the starting point to the point xy and n is the distance betw the point (x,y) and the endpoint.

internal division: (m/m+n) can also be used as a ratio, where (m+n) is the total length betwe the start to end point.

external division: (n/m-n) can also be used as a ratio, where (m-n) is the total length betwe the start to end point.

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12
Q

INTERNAL DIVISION: formula to get the coordiantes of (x,y)

A

x = (mX2+nX1)/m+n

y= (mY2+nY1)/(m+n)

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13
Q

EXTERNAL DIVISION: formula to get the coordiantes of (x,y)

A

x = (mX2-nX1)/m-n

y= (mY2-nY1)/(m-n)

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14
Q

in an internal division, if the ratio is (1:1) or the distance between both is equal. then the point is calleD?

A

the mid point

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15
Q

3 TYPES OF CONIC SECTION

A

ELLIPSE, PARABOLA, HYPERBOLA

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16
Q

IS A CIRCLE AN ELLIPSE?

A

IT IS A SPECIAL TYPE OF OF ELLIPSE

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17
Q

is a curve obtained as the intersection of the surface of a cone with a plane.

A

conic section

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18
Q

3 main parts of a CONIC SECTION

A

the focus- fix point inside the curve within the principal axis

principal axis- a line connecting the vertex and the focal point

directrix- a line outside the curve, perpendicular to the principal axis

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19
Q

In mathematics, _____ of a conic section is a non-negative real number that uniquely characterizes its shape. More formally two conic sections are similar if and only if they have the same _____.

____ is a measure of how much a conic section deviates from being circular.

ratio (f/d) where f is the distance of the point to the focus and d is the perpendicular distance to the directrix

A

ECCENTRICITY

- ONE WAY TO KNOW THE TYPE OF CONIC SECTION

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20
Q

4 WAYS TO KNOW THE TYPE OF CONIC SECTION

A 
BY CUTTING A CONE
BY THE ECCENTRICITY
BY GENERAL EQUATION
* CONIC DISCRIMINANT
* PRODUCT OF A & C
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21
Q

how to know the conic section by cutting a cone

A

imagine two cones facing each other forming an hourglass shape. where α is the angle of elevation with respect to the horizontal line in between the two cones.

if θ = 0, CIRCLE
if θ < α , ellipse
if θ = α , parabola
if θ > α , hyperbola

CEPH

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22
Q

how to know the conic section by eccentricity

A

e= f/d

each conic section has its own formula of eccentricity;
but if;
e=0 = circle
e < 1 = ellipse
e = 1  parabola
c > 1 = hyperbola

CEPH!

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23
Q

how to know the conic section by General eq (conic discriminant)

A

Ax^2+ Bxy+ Cy^2 + Dx + Ey + F = 0

  • APPLICABLE ONLY IS B≠0,
    THEN B^2 - 4AC.

ELLIPSE OR CIRCLE B^2 - 4AC <0
PARABOLA B^2 - 4AC =0
HYPER BOLA B^2 - 4AC > 0

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24
Q

how to know the conic section by General eq (product of A & C )

A

Ax^2+ Bxy+ Cy^2 + Dx + Ey + F = 0

  • APPLICABLE ONLY IS B=0,
    THEN (AC).
if ;
A = C circle
AC > 0 ellipse   (a & c have same sign)
AC = 0  parabola
AC < 0 hyperbola ( opposite sign/)
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25
Q

A is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.

A

CIRCLE

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26
Q

2 eq form of circle

A

GENERAL EQ
Ax^2+ Cy^2 + Dx + Ey + F = 0
A=C

STANDARD EQ
(x-h)^2 + (y-k)^2 = r^2
(h,k) coordinates of the center`

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27
Q

identify the center coordinates given the gen equation

A

since,
Ax^2+ Cy^2 + Dx + Ey + F = 0
& A=C
then Ax^2+ Ay^2 + Dx + Ey + F = 0

h= -D/2A

k= -E/2A

r= √(D^2 + E^2 - 4AF) / 2A

28
Q

5 PARTS OF A PARABOLA

A

AXIS OF SYMMETRY OR PRINCIPAL AXIS

VERTEX - intersection of conic section and principal axis (h,k)

FOCUS- where the conic section is constructed

LATUS RECTUM- A CHORD that passes through the focus, parallel to the directrix and has both end points on the curve

Directrix

29
Q

distance between the vertex and focal point

A

the focal length, (a)

30
Q

DISTANCE betwee vertex to the directrix

A

a, same with the distance between the vertex and focal point so.

31
Q

DISTANCE BETWEEN THE FOCAL POINT AND THE endpoints of the latus rectum.

A

2a on each sides,

so total length of latus rectum = 4a

32
Q

a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

A

parabola

33
Q

eccentricity value of parabola

A

1 = e

34
Q

orientation of the axis of symmetry of the parabola

A

since Ax^2+ Cy^2 + Dx + Ey + F = 0

if X is squared, then axis of symmetry is vertical

if y is squared, then the axis of symmetry is horizontal

35
Q

STANDARD EQ OF PARABOLA

A 
(x-y)= ±4a(y-k)
(y-k)=  ±4a(x-h)

a= focal length
+ =upward , facing right
- =downward, facing left

36
Q

latus rectum length

A

4a = LR

37
Q

10 PARTS OF AN ELLIPSE

A

2 latus rectum

2 directrix

2 focal point

2 vertices

principal axis

Major axis- longest diameter between endpoints within the ellipse ( vertex to vertex)

Minor axis- shortest diameter between endpoints within the ellipse & perpendicular to the major axis, passing through the center.

CENTER POINT= (h,k)

semi major axis = half of mmajor axis (a)

semi minor axis = half of minor axis (b)

focal length - distance between the center & a focal point (c)

38
Q

focal length for ellipse is the?

A

distance between the center & a focal point (c)

39
Q

all variables for ellipse to use.

A 
a = semi major
b = semi minor
c = focal length
D = distance between the two directrix
(h,k) = coordinates of center
40
Q

relation of semi major and semi minor axis in an equation

A

a^2 = b^2 + c^2

a= semi major
b=  semi minor
c= focal length
41
Q

ellipse general eq.

A

since Ax^2+ Bxy+ Cy^2 + Dx + Ey + F = 0

if Bxy=0, then
Ax^2+ Cy^2 + Dx + Ey + F = 0,
we know that the orientation of ellipse is either vertical or horizontal.

42
Q

standard eq of ellipse

A

[(x-h)^2]/a^2 + [(y-k)^2]/b^2 =1

a>b , a is gr8er since its major

43
Q

orientation of ellipse

A

given, [(x-h)^2]/a^2 + [(y-k)^2]/b^2 =1

if major-axis (a) is below the x coordinates, then principal axis is parallel or along the x axis.

44
Q

latus rectum of ELLIPSE

A

LR= 2b^2 /a

45
Q

eccentricity of a ellipse

A

e = a/ D = c/a

46
Q

unique line of a hyperbole, where it is a line that a curve approaches as it headed towards infinity.

A

asymptote

47
Q

major axis equivalent for the hyperbole, line from vertex to vertex of the conic section

A

TRANSERSE AXIS

48
Q

minor axis equivalent for the hyperbole, line that represent the width of the rectangular shape formed by the asymptote and vertices of the conic section hyperbola.

A

conjugate axis

  • perpendicular to the tranverse axis and principal axis,
  • passes through the center point
49
Q

how many dirextrix does a hyperbola have? 1?

A

2!

50
Q

four main distances in a hyperbola

A 
a= half a transverse axis, vertex to center distance
b= half a conjugate axis,  center to endpoint of rectangular shape
c= center to focus, focal length
D= distance between to directrix.
51
Q

in a hyperbola, a rectangular shape is formed with the center as its center, within it a right triangule shape can be seen. what are the sides of the triangle?

A

a^2 + b^2 = c^2

a= 1/2 transverse
b= 1/2 conjugate
c= focal length
52
Q

decribe the constant of a hyperbola which makes it in a shape of a hyperbola conic secion.

A

d1 - d2 =2a

the difference in distance between the two focal point of a hyperbola is always equal to the length between the vertices of each part of the hyperbola.`

53
Q

general eq of hyperbola.

A

GEN EQ OF HYPERBOLA:

since Ax^2+ Bxy+ Cy^2 + Dx + Ey + F = 0
principal axis is neither vert, or horiz.

discriminant mus be greater than zero to be
hyperbola.

Ax^2 - Cy^2 + Dx + Ey + F = 0
pricipal axis either vert or horiz.

54
Q

STANDARD EQ OF HYPERBOLA

A

[(x-h)^2]/a^2 - [(y-k)^2]/b^2 =1

a>b , a is gr8er since its transvers (kind of like major axis)
a is always positve.

55
Q

how to know the orientation of ahyperbola given
the standard eq.

[(x-h)^2]/a^2 - [(y-k)^2]/b^2 =1

A

if a is beneath the x coordinat, then the principal axis is parallel or along the x axis. (left to right)

if a is beneath the y coordinat, then the principal axis is parallel or along the y axis. (up & down)

56
Q

how to know the orientation of ahyperbola given
the gen eq

Ax^2 - Cy^2 + Dx + Ey + F = 0

A

Ax^2 - Cy^2 + Dx + Ey + F = 0
-Ax^2 + Cy^2 + Dx + Ey + F = 0

if A term is positive then, left to right
if C term is positive then, up and down

57
Q

latus rectum of hypherball-A

A

LR= 2b^2 /a

58
Q

Eccentricity of hyperbola

A

a/D = c/a = e

e > 1

59
Q

how to convert rec to pol and vice versa in calcu

A

use pol and rec function

use pol if u want to find the R - P
use Rec if u want to find the P- R

60
Q

how to convert rect to pol eq.

A

x^2 + y^2 = r^2

θ = tan^-1 (y/x)

61
Q

how to convert pol to req eq.

A 
y= rsinθ 
x= rcosθ
62
Q

describe the rectangular, & cylindrical coordinates for a 3 dimension point of view.

A

rectangulr (xy,z) where another variable is added

cylindrical is defined by the radius (ρ) ,angle of θ from x axis (2dimension angle), & the height or elevation (z)
(ρ,θ,z) form a cylinder when rotated upon the z axis.

63
Q

conversion between cyclindrican and rect.

A

RECT TO CYL.
x^2 + y^2 = ρ^2
θ = tan^-1 (y/x)
z=z

cyl to rect:
y= ρsinθ
x= ρcosθ

64
Q

how is spherical coordiante place

A

given 2 angles and 1 distance r
Φ - angle of point with respect to z axis
θ - angle of point with respect to x axis
r - distance of the point

(r,θ,Φ)

65
Q

convert rext to speherical and vice versa

A 
rect to spherical 
(x,y,z) to (r,θ,Φ)
θ =tan^-1(y/x)
r= √( x^2 + y^2+z^2)
Φ= cos^-1( z/√(x^2+y^2+z^2) )

spherical to rect:
x= rsinΦ cosθ
y= rsinΦ sinθ
z= rcosΦ

66
Q

translation and rotation of axis purpose? and formula

A

the process of replacing the axes in a Cartesian coordinate system with a new set of axes, parallel to the first, used to write equations of curves not centered about the origin.

translation:
x’= (x-h)
y’= (y-k)

roation axis :
x'=  xcosθ + ysinθ 
y'=  -xsinθ + ycosθ 
(+θ  = counterclockwise)
(-θ  = clockwise)
roation of point:
x'=  xcosθ + ysinθ 
y'=  -xsinθ + ycosθ 
(+θ  = clockwise)
(-θ  = counterclockwise)

(MATH) (11 decks)

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