Analytic Geom Flashcards
In general, quadratic equation, if the discriminant is zero, the curve is a figure that represent a/an
Parabola
Equations relating x and y that cannot readily be solved explicitly for y as a function of x or for x as a function of y. Such equations may nonetheless determine y as a function of x or vice versa, such function is called
Implicit function
In polar coordinates system, the length of the ray segment from a fixed origin is known as ____.
Radius vector
Given the equation 3x2 + 2x – 5y + 7 = 0. determine the curve.
Parabola
If eccentricity is less than one, then the curve is
Ellipse
Of what quadrant is A, if sec A is positive and csc A is negative?
IV
If the general equation of the conic is Ax2 + 2Bxy + Cy2 + Ey + F = 0 and B2 – 4AC > 0, then the conic is a/an
Hyperbola
What type of conic has equation of Ax2 + Cy2 + Dx + Ey + F = 0?
Ellipse
The graph of r = a + bcosθ is a
Limacon
In an ellipse, a chord which contains a focus and is in line perpendicular to the major axis is called
Latus rectum
If all the y-terms have even exponents, the curve is symmetric with respect to the ____.
X-axis
It can be defined as the set of all points in the plane the sum of whose distances from two fixed points is constant
Ellipse
If the equation is unchanged by the substitution of – x for x, its curve is symmetric with respect to the
Y-axis
What type of curve is generated by a point which moves in uniform circular motion about an axis, while travelling with a constant speed parallel to the axis?
Helix
It represents the distance of a point from the y-axis
Abscissa
A line passing through the focus and perpendicular to the directrix of a parabola is called
Axis of parabola
Locus of points on a side which rolls along a fixed line
Cycloid
If the product of the slopes of any two straight lines is negative 1, one of these is said to be ____ to the other.
Perpendicular
What is the curve represented by the equation r = aθ?
Spiral of Archimedes
Is the locus of a point that moves in a plane so that the difference of the distances from two fixed points of the locus is constant
Hyperbola
The length of the latus rectum of the parabola 𝑦 = 4𝑝𝑥2 is
1/4p
The cartesian or rectangular coordinates system was first introduced by
Descartes
Also known as the x-coordinate
Abscissa
The x-coordinate of a point is positive in what quadrants?
I and IV
The y-coordinate of a point is positive in what quadrants?
I and II
The rectangular coordinate system used to represent a complex number
Argand Diagram
A cartesian coordinates system in which the axes are not perpendicular
oblique coordinates system
The rectangular coordinate system in space is divided into eight compartments called
Octants
The angle of inclination of a straight line is the angle it makes with the
Positive x-axis
The points where the curve crossed the coordinates axes are called as the ___ with the axes
Intersections
A line which is perpendicular to the x-axis, has slope equal to
Infinity
If all the y-terms have even exponents, the curve is symmetric with respect to the
x-axis
If the equation is unchanged by the substitution of – x for x, and – y for y its curve is symmetric with respect to the
origin
If all of the terms of an equation have even exponents of if all of the terms have odd exponents, the curve is symmetric with respect to the
origin
If two linear equations, the x-coefficient of the first is equal to the y-coefficient of the send and the y-coefficient of the first is numerically equal but of opposite sign to the x-coefficient of the second, or vice- versa, the line represented are
perpendicular to each other
if two equations have the same line as their graph, the equations are said to be
dependent
In a linear equation Ax + By + C = 0, if B = 0 then the equation has the form of x = -C/A. This line is
parallel to the y-axis
A straight line where the curve approaches more and more closely but never touches it except at a limiting point of infinity
Asymptote
Who coined the word “asymptote”?
Thomas Hobbes
The path of a point which moves according to a given law or equation
Locus
The curve traced by a point moving in a plane is shown as the ____ of the point
Locus
A conic section is curve which is the intersection of
a cone and a plane
When the ellipse approaches a circle as a limiting shape, its eccentricity approaches
0
The set of points in a plane, the sum of whose distances form a fixed point is constant, is
circle
If a right circular cone is cut by a plane parallel to its base, it would reveal a/an
circle
A ___ to a circle is a line that has exactly one point in common with the circle.
tangent
A conic section whose eccentricity is always less than 1
ellipse
A locus of appoint which moves so that the sum of the distances from two fixed points (foci) is constant and equal to the length of the major axis.
ellipse
If the distance from the center to the focus of an ellipse is c, from the center to the vertex is a and from the center to the directrix is D, its eccentricity, is
c/a
A locus of point which moves so that it is always equidistant from a fixed point (focus) and from a fixed straight line (directrix)
parabola
The angle between the tangents at the end points of the latus rectum of a parabola is
90
The tangents of the parabola at the end points of its latus rectum intersect.
at the directrix
In general equation of a conic section Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, if A and C have different signs, then the curve is a/an
hyperbola
If the discriminant of a quadratic equation is greater than zero, the graph is a/an
hyperbola
A chord passing through the focus of a parabola and perpendicular to the axis of symmetry
latus rectum
The latus rectum of the parabola x2 = 4ay is
4a
If a and b are lengths of semi-major and semi-minor axis of an ellipse respectively, then what is the length of its latus rectum?
2b^2/2
The eccentricity of a regular hyperbola is
sqrt. of 2
The axis of the parabola that passes through the foci, vertices and center is called
transverse axis
The locus of a moving point in a plane so that the difference of its distance from two fixed points (foci) is constant
hyperbola
What is the term given to a circle with radius equal to half the transverse axis of the hyperbola or major axis of an ellipse and its center is the center of conics?
auxiliary circle
Which is NOT a central conic?
parabola
Confocal conics are conics
having the same foci
A line segment joining two of its points and passing through a focus of a conic
focal chord
Given the polar equation r = 3 / (1 + 3 cos θ). This is a graph of a/an
hyperbola
The equation r = 4 cos θ is a/an
circle
In polar coordinate system, the distance of any point P from the origin is called
radius vector
The plane curve traced out by a fixed point on the circle as the circle rolls along a line.
cycloid
A plane curve traced by a fixed point on a circle as it rolls along outside of a fixed circle.
epicycloid
A plane curve traced by a fixed point on a circle as it rolls along inside of a fixed circle.
hypocycloid
The equation x3 + y3 – 3ay = 0 represents a
folium of Descartes
Continuous curve traced by a point moving around fixed point in same plane are steadily increasing or decreasing distance
spiral
Locus of the ultimate intersections or curves in a system of curves.
envelope
Curve formed by uniform chain hanging freely from two points.
catenary
The locus of a point such that its radius vector is proportional to its vector angle.
Spiral of Archimedes
The graph of the equation r = acos 2θ is a
rosette
The locus of a point which rolls on a straight line (x-axis)
trochoid
The equation r = a(1 +cos θ) is a polar equation of
cardioids
The equation r^2 = a^2 cos θ is a
lemniscate
The equation r = a cos θ is a
rosette
The equation r – aθ =0 is a
spiral
The equation r = a cos θ + b is a
limacon
The equation r = a (secθ – tan θ) is a
strophoid
The equation r = a (4cosθ – secθ) is a
trisectrix
The equation (x^2 – 2ay – a^2)^2 = y^2 (a^2 – x^2) is a
cocked hat
The equation x^2 + y^2 = a^2 is a
lames quartic
The equation ax^2 = y^2 (2a – y) is the equation of
the top
The equation (x^2 + y^2)^2 = ax^2 y is an equation of
bifolium
The equation y^2 = (x^2 + 1)^2 (2 – x^2)^3 is an equation of
fish mouth
A curve or surface that is tangential to each of the family of curves or surfaces
envelope
A curve that describes the locus of the centers of curvatures of another curve to which its tangent is normal
evolute
____ is formed by intersection of rays from the point reflected or refracted from a curve surface.
caustic
