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Bryx_Math > Analytic Geometry > Flashcards

Analytic Geometry Flashcards

1
Q

The x-coordinate, measured from the y-axis

A

Abscissa

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2
Q

The y-coordinate, measured from the x-axis

A

Ordinate

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3
Q

Distance between two points formula

A

d=sqrt((x2-x1)^2+(y2-y1)^2)

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4
Q

Distance between two points in a space formula

A

d=sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1))

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5
Q

Slope of a line formula

A

m=(y2-y1)/(x2-x1)

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6
Q

relationship of the slopes of perpendicular lines

A

m2=-1/m1 (negative reciprocal)

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7
Q

Angle between two lines formula

A

theta=tan^-1((m2-m1)/(1+m1m2))

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8
Q

Distance between a point and a line formula

A

d=(Ax1+By1+C)/(+/-sqrt(A^2+B^2))
point(x1,y1)
line equation Ax+By+C=0

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9
Q

Distance between two parallel lines formula

A

d=(C1-C2)/(sqrt(A^2+B^2))

where C1 and C2 are the constants in the general equation of a line

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10
Q

Area of a polygon by coordinates

A

A=(1/2)*((x1y2+x2y3+x3y1)-(y1x2+y2x3+y3x1))

basically just get the determinant, but the points should be arranged in a cyclic manner

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11
Q

General Equation of a line

A

Ax+By+C=0

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12
Q

Point slope form of a line

A

y-y1=m(x-x1)

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13
Q

Slope-Intercept form

A

y=mx+b

where b=y-intercept

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14
Q

Two-point form

A

y-y1=((y2-y1)/(x2-x1))*(x-x1)

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15
Q

Intercept form

A

x/a+y/b=1
a=x-intercept
b=y-intercept

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16
Q

Who introduced the term conic? he is a renowned mathematician and astronomer

A

Apollonius

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17
Q

A conic produced when the cutting plane is parallel to the base of the cone

A

Circle

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18
Q

A conic produced when the cutting plane is not parallel(or inclined) to the base of the cone

A

Ellipse

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19
Q

A conic produced when the cutting plane is parallel to the element(or generatrix) of the cone

A

Parabola

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20
Q

A conic produced when the cutting plane is parallel to the axis of the cone

A

Hyperbola

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21
Q

General equation of Conic Section

A

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

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22
Q

In the general equation of conic section, if B is not equal to zero, the curve can be determined by the value of the determinant___________

A

B^2-4AC

*can only be used if B=/0

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23
Q

If the determinant of the conic section is <0, what is the conic?

A

Ellipse

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24
Q

If the determinant of the conic section is =0, what is the conic?

A

Parabola

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25
Q

If the determinant of the conic section is >0, what is the conic?

A

Hyperbola

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26
Q

What is the eccentricity of an Ellipse

A

<1.0

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27
Q

What is the eccentricity of a Parabola

A

=1.0

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28
Q

What is the eccentricity of an Hyperbola

A

> 1.0

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29
Q

In the general equation of a Conic, if B=0 what is the conic if A=C?

A

Circle

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30
Q

In the general equation of a Conic, if B=0 what is the conic if A=/C but have the same sign

A

Ellipse

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31
Q

In the general equation of a Conic, if B=0 what is the conic if A and C have different signs

A

Hyperbola

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32
Q

In the general equation of a Conic, if B=0 what is the conic if either A or C is zero

A

Parabola

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33
Q

A circle reflects rays issued from the focus back to the ______ of the circle

A

center

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34
Q

Parabola reflects rays issued from the focus as _______ outgoing beam

A

parallel (with respect to its axis)

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35
Q

Ellipse reflects rays issued from the focus into__________

A

the other focus

36
Q

Hyperbola reflects rays issued from the focus as if _____________

A

Coming from the other focus

37
Q

General Equation of a circle

A

x^2+y^2+Dx+Ey+F=0

38
Q

Standard equation of a circle

A 
x^2+y^2=r^2
at center (h,k):
 (x-h)^2+(y-k)^2=r^2
39
Q

Center of a circle given the general form

A

h=-D/2A k=-E/2A

40
Q

Radius of a circle given the general form

A

r=sqrt((D^2+E^2-4AF)/(4A^2))

41
Q

Equation of a line tangent to a circle with vertex at origin

A

y=(-x1*x/y1)+r^2/y1

42
Q

slope of a line tangent to a circle located at origin

A

m=-x1/y1

43
Q

It is a locus of all points which moves so that it is always equidistant to a fixed point called focus and a fixed straight line called directrix

A

Parabola

44
Q

General equations of a parabola

A

axis parallel to y-axis: Ax^2+Dx+Ey+F=0

axis parallel to x-axis: Cy^2+Dx+Ey+F=0

45
Q

Standard equation of parabola

A

(y-k)^2=4a(x-h) (opens right)
(y-k)^2=-4a(x-h) (opens left)
(x-h)^2=4a(y-k) (Opens up)
(x-h)^2=-4a(y-k) (Opens down)

46
Q

It is the ratio of the distance to the focus to the distance to the directrix

A

eccentricity

e=f/d

47
Q

Latus rectum formula of parabola

A

LR=4a

48
Q

equation of a line tangent to a parabola (vertex @ (0,0)) at a given point (X1,Y1)

A

y = (2ax / (Y1)) + (2a(X1) / (Y1))

49
Q

Slope of a line tangent to a parabola (vertex @ (0,0)) at a given point (X1,Y1)

A

m = 2a / Y1

50
Q

A locus of a point which moves so that the sum of its distance to the fixed points (foci) is Constant

A

Ellipse

51
Q

The sum of the distance from a point of an ellipse to the two foci is equal to?

A

twice the length of the major axis (2a)

52
Q

Ellipse General Equation?

A

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

53
Q

the distance between the two vertices of the ellipse

A

Twice the length of the major axis (2a)

54
Q

formula that relates “c” to major(a) and minor(b) axis in an ellipse

A

a^2 = b^2 + c^2

55
Q

Ellipse Intercept Form Formula

A

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

56
Q

How do we determine if ellipse’s major axis is vertical or horizontal WRT the X-Axis?

A

Vertical if:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1

Horizontal if:
((x - h)^2 / b^2) + ((y - k)^2 / a^2) = 1

57
Q

Alternate Eccentricity formula for ellipse

A

e = a / D
D - Directrix-to-center distance

   OR

e = c / a

58
Q

value of Ellipse’s Eccentricity is ,= 1?

A

e(ellipse) < 1

59
Q

Latus rectum of an ellipse

A

LR = 2b^2/a

60
Q

Coordinates of ellipse’s center Given the General Equation

A 
h = -D / 2A
k = -E /2C
61
Q

A Locus of a point which moves so that the difference of the distances to the two foci is constant

A

Hyperbola

62
Q

It is the axis that passes through the foci, vertices and the center of the hyperbola

A

Transverse Axis

63
Q

It is the axis which is perpendicular to the transverse axis

A

Conjugate Axis

64
Q

Formula for the length of the transverse axis of a hyperbola when the transverse axis is horizontal

A

2a or 2sqrt(C)

where C is the absolute value of the coefficient of y^2

65
Q

Formula for the length of the conjugate axis of a hyperbola when the transverse axis is horizontal

A

2b or 2sqrt(A)

where A is the absolute value of the coefficient of x^2

66
Q

The relationship of a, b, and c in a hyperbola?

A

a^2 + b^2 = c^2

67
Q

The General equation for a hyperbola having a horizontal transverse axis

A

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

68
Q

The General equation for the hyperbola having a vertical transverse axis

A

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

69
Q

Formula for the length of the transverse axis of a hyperbola when the transverse axis is vertical

A

2a or 2sqrt(A)

where A is the absolute value coefficient of x^2

70
Q

Formula for the length of the conjugate axis of a hyperbola when the transverse axis is vertical

A

2b or 2sqrt(C)

where C is the absolute value coefficient of y^2

71
Q

Standard equation of a hyperbola having a horizontal transverse axis and a center at the origin

A

(x^2)/(a^2)-(y^2)/(b^2)=1

72
Q

Standard equation of a hyperbola having a vertical transverse axis and a center at the origin

A

(y^2)/(a^2)-(x^2)/(b^2)=1

73
Q

Standard equation of a hyperbola having a horizontal transverse axis and a center at the (h,k)

A

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

74
Q

Standard equation of a hyperbola having a vertical transverse axis and a center at the (h,k)

A

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

75
Q

Value of eccentricity of a hyperbola

A

e>1

76
Q

Formula for the eccentricity of a hyperbola

A

e=c/a or a/D

77
Q

Formula for the Latus Rectum of a hyperbola

A 
LR = (2b^2)/a
a= semi-major axis
b= semi-minor axis
78
Q

Formulas for the center of a hyperbola

A 
h = -D/2A
K= -E/2C
79
Q

It is the coordinates of a point where the position of a point is determined by the length of the segment and the angle of the ray

A

Polar Coordinates

80
Q

Other names for polar angle

A

Vectoral Angle
Argument
Amplitude
Azimuth

81
Q

Relationship between polar angle and rectangular coordinates

A

x=r᛫cos(theta)
y=r᛫sin(theta)
r=√(x^2+y^2)

82
Q

distance between a line and a plane is space

A

d=(ax1+by1+cz1+d)/(√(a²+b²+c²))

83
Q

formula for the area under the first octant for the plane x/a+y/b+z/c=1

A

Area=1/2(√(a²b²+b²c²+a²c²))

84
Q

formula for the volume of the region in the first octant bounded by the plane x/a+y/b+z/c=1

A

Volume=abc/6

85
Q

formula for the centroid of the region in the first octant bounded by the plane x/a+y/b+z/c=1

A

x=a/4 y=b/4 z=c/4

86
Q

formula for Surface area of sphere of radius r in the first octant

A

Surface Area=πr²/2

Bryx_Math (9 decks)

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