Conic sections are obtained by the intersection of the surface of a cone with a plane, and have certain features.
Describe the parts of a conic section and how conic sections can be thought of as cross-sections of a double-cone
- A conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane; the three types are parabolas, ellipses, and hyperbolas.
- A conic section can be graphed on a coordinate plane.
- Every conic section has certain features, including at least one focus and directrix. Parabolas have one focus and directrix, while ellipses and hyperbolas have two of each.
- A conic section is the set of points
PPP
whosedistance to the focus is a constant multiple of the distance from
PPP
to the directrix of the conic.
- vertex: An extreme point on a conic section.
- asymptote: A straight line which a curve approaches arbitrarily closely as it goes to infinity.
- locus: The set of all points whose coordinates satisfy a given equation or condition.
- focus: A point used to construct and define a conic section, at which rays reflected from the curve converge (plural: foci).
- nappe: One half of a double cone.
- conic section: Any curve formed by the intersection of a plane with a cone of two nappes.
- directrix: A line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two (plural: directrices).
Conic sections are generated by the intersection of a plane with a cone. If the plane is parallel to the axis of revolution (the
yyy
-axis), then the conic section is a hyperbola. If the plane is parallel to the generating line, the conic section is a parabola. If the plane is perpendicular to the axis of revolution, the conic section is a circle. If the plane intersects one nappe at an angle to the axis (other than90∘90^{\circ}90∘
), then the conic section is an ellipse.These properties that the conic sections share are often presented as the following definition, which will be developed further in the following section. A conic section is the locus of points
PPP
whose distance to the focus is a constant multiple of the distance fromPPP
to the directrix of the conic. These distances are displayed as orange lines for each conic section in the following diagram.Discuss how the eccentricity of a conic section describes its behavior
- Eccentricity is a parameter associated with every conic section, and can be thought
of as a measure of how much the conic section deviates from being circular.
- The eccentricity of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix.
- The value of
eee
can be used to determine the type of conic section. Ife=1e= 1e=1
it is a parabola, ife<1e < 1e<1
it is an ellipse, and ife>1e > 1e>1
it is a hyperbola.
- eccentricity: A parameter of a conic section that describes how much the conic section deviates from being circular.
eee
, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.The eccentricity of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix. The value of
eee
is constant for any conic section. This property can be used as a general definition for conic sections. The value ofeee
can be used to determine the type of conic section as well:- If
e=1e = 1e=1
, the conic is a parabola - If
e<1e < 1e<1
, it is an ellipse - If
e>1e > 1e>1
, it is a hyperbola
111
.For an ellipse, the eccentricity is less than
111
. This means that, in the ratio that defines eccentricity, the numerator is less than the denominator. In other words, the distance between a point on a conic section and its focus is less than the distance between that point and the nearest directrix.Conversely, the eccentricity of a hyperbola is greater than
111
Discuss the properties of different types of conic sections
- Conic sections are a particular type of shape formed by the intersection of a plane and a right circular cone. Depending on the angle between the plane and the cone, four different intersection shapes can be formed.
- The types of conic sections are circles, ellipses, hyperbolas, and parabolas.
- Each conic section also has a degenerate form; these take the form of points and lines.
- degenerate: A conic section which does not fit the standard form of equation.
- asymptote: A line which a curved function or shape approaches but never touches.
- hyperbola: The conic section formed by the plane being perpendicular to the base of the cone.
- focus: A point away from a curved line, around which the curve bends.
- circle: The conic section formed by the plane being parallel to the base of the cone.
- ellipse: The conic section formed by the plane being at an angle to the base of the cone.
- eccentricity: A dimensionless parameter characterizing the shape of a conic section.
- Parabola: The conic section formed by the plane being parallel to the cone.
- vertex: The turning point of a curved shape.
eee
. The four conic section shapes each have different values ofeee
.- A vertex, which is the point at which the curve turns around
- A focus, which is a point not on the curve about which the curve bends
- An axis of symmetry, which is a line connecting the vertex and the focus which divides the parabola into two equal halves
All parabolas possess an eccentricity value
e=1e=1e=1
. As a direct result of having the same eccentricity, all parabolas are similar, meaning that any parabola can be transformed into any other with a change of position and scaling. The degenerate case of a parabola is when the plane just barely touches the outside surface of the cone, meaning that it is tangent to the cone. This creates a straight line intersection out of the cone's diagonal. Non-degenerate parabolas can be represented with quadratic functions such asf(x)=x2f(x) = x^2f(x)=x2
A circle is formed when the plane is parallel to the base of the cone. Its intersection with the cone is therefore a set of points equidistant from a common point (the central axis of the cone), which meets the definition of a circle. All circles have certain features:- A center point
- A radius, which the distance from any point on the circle to the center point
All circles have an eccentricity
e=0e=0e=0
. Thus, like the parabola, all circles are similar and can be transformed into one another. On a coordinate plane, the general form of the equation of the circle is(x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2(x−h)2+(y−k)2=r2
where
(h,k)(h,k)(h,k)
are the coordinates of the center of the circle, andrrr
is the radius. The degenerate form of the circle occurs when the plane only intersects the very tip of the cone. This is a single point intersection, or equivalently a circle of zero radius.0.50.50.5
, a parabola in green with the required eccentricity of111
, and a hyperbola in blue with an example eccentricity of222
. It also shows one of the degenerate hyperbola cases, the straight black line, corresponding to infinite eccentricity. The circle is on the inside of the parabola, which is on the inside of one side of the hyperbola, which has the horizontal line below it. In this way, increasing eccentricity can be identified with a kind of unfolding or opening up of the conic section.- A major axis, which is the longest width across the ellipse
- A minor axis, which is the shortest width across the ellipse
- A center, which is the intersection of the two axes
- Two focal points —for any point on the ellipse, the sum of the distances to both focal points is a constant
Ellipses can have a range of eccentricity values:
0≤e<10 \leq e < 10≤e<1
. Notice that the value000
is included (a circle), but the value111
is not included (that would be a parabola). Since there is a range of eccentricity values, not all ellipses are similar. The general form of the equation of an ellipse with major axis parallel to the x-axis is:(x−h)2a2+(y−k)2b2=1\displaystyle{ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 }a2(x−h)2+b2(y−k)2=1
where
(h,k)(h,k)(h,k)
are the coordinates of the center,2a2a2a
is the length of the major axis, and2b2b2b
is the length of the minor axis. If the ellipse has a vertical major axis, theaaa
andbbb
labels will switch places. The degenerate form of an ellipse is a point, or circle of zero radius, just as it was for the circle. A hyperbola is formed when the plane is parallel to the cone's central axis, meaning it intersects both parts of the double cone. Hyperbolas have two branches, as well as these features:- Asymptote lines—these are two linear graphs that the curve of the hyperbola approaches, but never touches
- A center, which is the intersection of the asymptotes
- Two focal points, around which each of the two branches bend
- Two vertices, one for each branch
(x−h)2a2−(y−k)2b2=1\displaystyle{ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 }a2(x−h)2−b2(y−k)2=1
where
(h,k)(h,k)(h,k)
are the coordinates of the center. Unlike an ellipse,aaa
is not necessarily the larger axis number. It is the axis length connecting the two vertices.The eccentricity of a hyperbola is restricted to
e>1e > 1e>1
, and has no upper bound. If the eccentricity is allowed to go to the limit of+∞+\infty+∞
(positive infinity), the hyperbola becomes one of its degenerate cases—a straight line. The other degenerate case for a hyperbola is to become its two straight-line asymptotes. This happens when the plane intersects the apex of the double cone.