ANALYTIC
GEOMETRY
CONIC SECTIONS
GENERAL DEFINITION OF CONIC
SECTION
locus (or path) of a point that moves
such that the ratio of its distance from a
fixed point (focus) and a fixed line
(directrix) is constant. This constant
ratio is called the eccentricity of the
conic.
Eccentricity of a conic
𝒇𝟏
𝒇𝟐
𝒇𝟑
𝒆=
=
=
𝒅𝟏
𝒅𝟐
𝒅𝟑
If 𝑒 = 0, it’s a circle.
If 𝑒 = 1, it’s a parabola.
If 𝑒 < 1, it’s an ellipse.
If 𝑒 > 1, it’s a hyperbola.
CIRCLE
GENERAL DEFINITION
A locus of a point which moves at a
constant distance from a fixed point
called center and the constant distance of
any point from the center is called the
radius.
Center at 𝐶(ℎ, 𝑘)
𝒙 − 𝒉 𝟐 + 𝒚 − 𝒌 𝟐 = 𝒓𝟐
Center at origin
𝒙 𝟐 + 𝒚𝟐 = 𝒓 𝟐
General form
𝒙𝟐 + 𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎
Center: ( h , k )
h = - D/2
k = - E/2
PARABOLA
GENERAL DEFINITION
A locus of a point that moves such that its
distance from a fixed point called the
focus is always equal to its distance from
a fixed line called the directrix
Standard Equation
( x – h )2 = 4a ( y – k )
( x – h )2 = -4a ( y – k )
( y – k )2 = 4a ( x – h )
( y – k )2 = -4a ( x – h )
Upward
Downward
to the right
to the left
Note: (ℎ, 𝑘) is the vertex of the parabola.
ELLIPSE
GENERAL DEFINITION
A locus of a point that moves such that
the sum of its distances from two fixed
points called the foci is constant.
Standard Equation
Properties of Ellipse
HYPERBOLA
GENERAL DEFINITION
A locus of a point that moves such that the difference of its distance between two fixed
points called the foci is constant.
POINTS AND LINES
ANGLES BETWEEN TWO LINES
_
𝜽 = 𝒕𝒂𝒏 𝟏
𝒎 𝟐 − 𝒎𝟏
𝟏 + 𝒎 𝟏𝒎 𝟐
Or
_
_
𝜽 = 𝒕𝒂𝒏 𝟏 𝒎𝟐 − 𝒕𝒂𝒏 𝟏 𝒎𝟏
Note:
1. Two lines are parallel if their slopes are equal.
2. Two lines are perpendicular if the product of their slopes is -1.
LINE AND POINT DISTANCE
Distance from (𝑥1, 𝑦1) to 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
𝑑=
𝐴𝑥1 + 𝐵𝑦1 + 𝐶
𝐴2 + 𝐵2
Distance from (𝑥1, 𝑦1) to 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
𝑑=
𝐶2 − 𝐶1
𝐴2 + 𝐵2
POLAR COORDINATE SYSTEM
In this system, the location of a point is
expressed by its distance 𝑟 from a fixed
point called the pole and its angle θ from a
fixed line, usually the +𝑥-axis.
Relationship between Polar and Cartesian
Coordinate Systems:
𝑟2 = 𝑥2 + 𝑦2 𝑜𝑟 𝑟 =
𝑥2 + 𝑦 2
𝑥 = 𝑟𝑐𝑜𝑠𝜃 , 𝑦 = 𝑟𝑠𝑖𝑛𝜃
𝑡𝑎𝑛𝜃 =
𝑦
=𝑚
𝑥