Analytic Geometry Flashcards
The x-coordinate, measured from the y-axis
Abscissa
The y-coordinate, measured from the x-axis
Ordinate
Distance between two points formula
d=sqrt((x2-x1)^2+(y2-y1)^2)
Distance between two points in a space formula
d=sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1))
Slope of a line formula
m=(y2-y1)/(x2-x1)
relationship of the slopes of perpendicular lines
m2=-1/m1 (negative reciprocal)
Angle between two lines formula
theta=tan^-1((m2-m1)/(1+m1m2))
Distance between a point and a line formula
d=(Ax1+By1+C)/(+/-sqrt(A^2+B^2))
point(x1,y1)
line equation Ax+By+C=0
Distance between two parallel lines formula
d=(C1-C2)/(sqrt(A^2+B^2))
where C1 and C2 are the constants in the general equation of a line
Area of a polygon by coordinates
A=(1/2)*((x1y2+x2y3+x3y1)-(y1x2+y2x3+y3x1))
basically just get the determinant, but the points should be arranged in a cyclic manner
General Equation of a line
Ax+By+C=0
Point slope form of a line
y-y1=m(x-x1)
Slope-Intercept form
y=mx+b
where b=y-intercept
Two-point form
y-y1=((y2-y1)/(x2-x1))*(x-x1)
Intercept form
x/a+y/b=1
a=x-intercept
b=y-intercept
Who introduced the term conic? he is a renowned mathematician and astronomer
Apollonius
A conic produced when the cutting plane is parallel to the base of the cone
Circle
A conic produced when the cutting plane is not parallel(or inclined) to the base of the cone
Ellipse
A conic produced when the cutting plane is parallel to the element(or generatrix) of the cone
Parabola
A conic produced when the cutting plane is parallel to the axis of the cone
Hyperbola
General equation of Conic Section
Ax^2+Bxy+Cy^2+Ey+F=0
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___________
B^2-4AC
*can only be used if B=/0
If the determinant of the conic section is <0, what is the conic?
Ellipse
If the determinant of the conic section is =0, what is the conic?
Parabola
If the determinant of the conic section is >0, what is the conic?
Hyperbola
What is the eccentricity of an Ellipse
<1.0
What is the eccentricity of a Parabola
=1.0
What is the eccentricity of an Hyperbola
> 1.0
In the general equation of a Conic, if B=0 what is the conic if A=C?
Circle
In the general equation of a Conic, if B=0 what is the conic if A=/C but have the same sign
Ellipse
In the general equation of a Conic, if B=0 what is the conic if A and C have different signs
Hyperbola
In the general equation of a Conic, if B=0 what is the conic if either A or C is zero
Parabola
A circle reflects rays issued from the focus back to the ______ of the circle
center
Parabola reflects rays issued from the focus as _______ outgoing beam
parallel (with respect to its axis)
Ellipse reflects rays issued from the focus into__________
the other focus
Hyperbola reflects rays issued from the focus as if _____________
Coming from the other focus
General Equation of a circle
x^2+y^2+Dx+Ey+F=0
Standard equation of a circle
x^2+y^2=r^2 at center (h,k): (x-h)^2+(y-k)^2=r^2
Center of a circle given the general form
h=-D/2A k=-E/2A
Radius of a circle given the general form
r=sqrt((D^2+E^2-4AF)/(4A^2))
Equation of a line tangent to a circle with vertex at origin
y=(-x1*x/y1)+r^2/y1
slope of a line tangent to a circle located at origin
m=-x1/y1
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
Parabola
General equations of a parabola
axis parallel to y-axis: Ax^2+Dx+Ey+F=0
axis parallel to x-axis: Cy^2+Dx+Ey+F=0
Standard equation of parabola
(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)
It is the ratio of the distance to the focus to the distance to the directrix
eccentricity
e=f/d
Latus rectum formula of parabola
LR=4a
equation of a line tangent to a parabola (vertex @ (0,0)) at a given point (X1,Y1)
y = (2ax / (Y1)) + (2a(X1) / (Y1))
Slope of a line tangent to a parabola (vertex @ (0,0)) at a given point (X1,Y1)
m = 2a / Y1
A locus of a point which moves so that the sum of its distance to the fixed points (foci) is Constant
Ellipse
The sum of the distance from a point of an ellipse to the two foci is equal to?
twice the length of the major axis (2a)
Ellipse General Equation?
Ax^2 + Cy^2 + Dx + Ey +F = 0
the distance between the two vertices of the ellipse
Twice the length of the major axis (2a)
formula that relates “c” to major(a) and minor(b) axis in an ellipse
a^2 = b^2 + c^2
Ellipse Intercept Form Formula
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
How do we determine if ellipse’s major axis is vertical or horizontal WRT the X-Axis?
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
Alternate Eccentricity formula for ellipse
e = a / D
D - Directrix-to-center distance
OR
e = c / a
value of Ellipse’s Eccentricity is ,= 1?
e(ellipse) < 1
Latus rectum of an ellipse
LR = 2b^2/a
Coordinates of ellipse’s center Given the General Equation
h = -D / 2A k = -E /2C
A Locus of a point which moves so that the difference of the distances to the two foci is constant
Hyperbola
It is the axis that passes through the foci, vertices and the center of the hyperbola
Transverse Axis
It is the axis which is perpendicular to the transverse axis
Conjugate Axis
Formula for the length of the transverse axis of a hyperbola when the transverse axis is horizontal
2a or 2sqrt(C)
where C is the absolute value of the coefficient of y^2
Formula for the length of the conjugate axis of a hyperbola when the transverse axis is horizontal
2b or 2sqrt(A)
where A is the absolute value of the coefficient of x^2
The relationship of a, b, and c in a hyperbola?
a^2 + b^2 = c^2
The General equation for a hyperbola having a horizontal transverse axis
Ax^2-Cy^2+Dx+Ey+F=0
The General equation for the hyperbola having a vertical transverse axis
Cy^2-Ax^2+Dx +Ey+F=0
Formula for the length of the transverse axis of a hyperbola when the transverse axis is vertical
2a or 2sqrt(A)
where A is the absolute value coefficient of x^2
Formula for the length of the conjugate axis of a hyperbola when the transverse axis is vertical
2b or 2sqrt(C)
where C is the absolute value coefficient of y^2
Standard equation of a hyperbola having a horizontal transverse axis and a center at the origin
(x^2)/(a^2)-(y^2)/(b^2)=1
Standard equation of a hyperbola having a vertical transverse axis and a center at the origin
(y^2)/(a^2)-(x^2)/(b^2)=1
Standard equation of a hyperbola having a horizontal transverse axis and a center at the (h,k)
(x-h)^2/(a^2)-(y-k)^2/(b^2)=1
Standard equation of a hyperbola having a vertical transverse axis and a center at the (h,k)
(y-k)^2/(a^2)-(x-h)^2/(b^2)=1
Value of eccentricity of a hyperbola
e>1
Formula for the eccentricity of a hyperbola
e=c/a or a/D
Formula for the Latus Rectum of a hyperbola
LR = (2b^2)/a a= semi-major axis b= semi-minor axis
Formulas for the center of a hyperbola
h = -D/2A K= -E/2C
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
Polar Coordinates
Other names for polar angle
Vectoral Angle
Argument
Amplitude
Azimuth
Relationship between polar angle and rectangular coordinates
x=r᛫cos(theta)
y=r᛫sin(theta)
r=√(x^2+y^2)
distance between a line and a plane is space
d=(ax1+by1+cz1+d)/(√(a²+b²+c²))
formula for the area under the first octant for the plane x/a+y/b+z/c=1
Area=1/2(√(a²b²+b²c²+a²c²))
formula for the volume of the region in the first octant bounded by the plane x/a+y/b+z/c=1
Volume=abc/6
formula for the centroid of the region in the first octant bounded by the plane x/a+y/b+z/c=1
x=a/4 y=b/4 z=c/4
formula for Surface area of sphere of radius r in the first octant
Surface Area=πr²/2
