Analytic Geometry Flashcards
The angle that the line makes with the X axis is the angle of inclination of the line, which is equal to the __________.
Tangent of the slope (tan theta = slope)
What is translated axis?
origin is relocated
How to get the point with respect to the translated axis?
(x-xt, y-yt)
note: and origin point sa translated axis ang i minus
Defined by two perpendicular axis
cartesian plane
Who invented the cartesian plane
Rene Descartes
Distance between two points formula
d= sqrt [(x1-x2)^2 + (y1-y2)^2]
Division of Line Segment
R= AP/Ab
R= (x-x1)/(x2-x1)
or
R= (y-y1)/(y2-y1)
Midpoint
Average of the x and y coordinates
Centroid of a triang;e
Average of the coordinates of the vertices
Area of Polygon (coordinate method)
A=1/2 (summation of xy)
Mode 32
x y
x1 y2-y3
x2 y3-y1
x3 y1-y2
(Present = Future - Past)
AC > apps > S-Sum
Equation of Lines General Form
Ax+By+C=0
Equation of Lines Standard Form
Ax+By=C
Slope Intercept Form
y= mx+b
Point Slope Form
m = (y-y1)/(x-x1)
Two point of form
(y2-y1)/(x2-x1) = (y-y1)/(x-x1)
Intercept Form
x/a +y/b = 1
Parallel Slope Line Equation
m1=m2
Perpendicular Slope Line Equation
m2= -1/m1
Angle between 2 lines
tan theta = (m2-m1) / 1+m1m2
Distance betweern a point and a line
d= abs [Ax+By+C / sqrt (A^2+B^2)]
Distance between 2 lines
d= abs [C1-C2 / sqrt (A^2+B^2)]
note: A and B of two lines should have the same coefficient
a set of all points whose location satisfies or is determined by one or more specific conditions
Locus of Points
Determining Conic Section: Circle
A=C same sign
Determining Conic Section: Ellipse
A is not equal to C but has same sign
Determing conic section: parabola
Only one (x or y) is squared
Determing conic section: Hyperbola
A and C not same sign
Circle General Fomula
x^2+y^2+dx+ey+f=0
Circle standard formula
(x-h)^2 + (y-k)^2 = r^2
Locus of point equidistant from a fixed point call focus and a fixed line call directrix
parabola
How to determine which axis is parabola parallel to
the term with 4a in standard from or whichever among x and y is not squared
if 4a > 0, parabolaā¦
opens upward or to the right
if 4a < 0, parabolaā¦
opens downward or to the left
Parabola eccentricity
always equal to 1
Parabola distance: vertex to focus
a
Parabola distance: vertex to directrix
a
Parabola distance: focus to vertex
a
Parabola distance: Latus rectum
4a
What is axis of symmetry?
line that passes through vertex and focus
Parabola standard formula
4a(y-k)=(x-h)^2
or
4a(x-h)=(y-k)^2
Parabola General Formula
Ax^2 + dx + ey + f = 0
or
Cy^2 + dx + ey + f = 0
Ellipse general formula
Ax^2 + Cy^2 + dx + ey + f =0
(same sign but not same value)
Ellipse standard formula
(x-h)^2/a^2 + (y-k)^2/b^2 =1
notes
a is always greater than b. kung asa si a. didto ang major axis (orientation of ellipse)
Ellipse property:
a^2 = b^2 + c^2
Eccentricity
e = c/a
or
e = a-c / d-a
Ellipse distance: center to directix
d
d= a/e
Ellipse distance: center to focus
c
Ellipse distance: center to vertex
a
Ellipse distance: center to covertex
b
Ellipse distance: major axis
2a
Ellipse distance: minor axis
2b
Eccentricity eā
eā= c/b
Ellipse distance: Latus Rectum
LR= 2b^2/ a
Ellipse distance: Flatness
f= a-b / a
Hyperbola general form
Ax^2 + Cy^2 + dx + ey + f =0
(different sign and value)
Hyperbola standard form
(x-h)^2/a^2 - (y-k)^2/b^2 =1
note: a is always located a the positive term and a dictates where the hyperbola is parallel with
Hyperbola property
c^2 = a^2 + b^2
eccentricity of hyperbola
e= c/a
e > 1.0
Transverse Axis parallel to x-axis: Asymptote eqn
y-k = +/- b/a (x-h)
Transverse Axis parallel to y-axis: Asymptote eqn
y-k = +/- a/b (x-h)
What axis is this parabola parallel with: (x-h)^2/a^2 - (y-k)^2/b^2 =1
x-axis
What axis is this hyperbola parallel with: (y-k)^2/a^2 - (x-h)^2/b^2 =1
y-axis
What is transverse axis?
the longer axis of the ellipse
What is conjugate axis?
the shorter axis of the ellipse
Polar Coordinates r^2 equation
r^2 = x^2 + y^2
Polar Coordinates: x =
r cos theta
Polar Coordinates: y =
r sin theta
Polar Coordinates: theta is equal to
arctan (y/x)
Polar Coordinates: Caltech rec to pol
Shift +
Polar Coordinates: Caltech pol to rec
Shift -
Polar Coordinates: Caltech getting distance
mode 2 (complex)
|x2-x1, < theta 2 - theta 1 |
It is a pattern of point that describes its relation to distance of a point (x,y) from one fixed entity to that of another entity.
Locus
What is the slope of the horizontal line
0
Another term for slope
Inclination
