+118058 Melody
moderator
+36444 ElectricPavlov
+33217 Alan
moderator
+14044 asinus
moderator
+3867 Probolobo
+2448 BuilderBoi
+2423 GingerAle
+1240 Lightning
+1155 nerdiest
+941 proyaop
+873 MathProblemSolver101
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
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. A cone and conic sections: The nappes and the four conic sections. Each conic is determined by the angle the plane makes with the axis of the cone. While each type of conic section looks very different, they have some features in common. For example, each type has at least one focus and directrix. A focus is a point about which the conic section is constructed. In other words, it is a point about which rays reflected from the curve converge. A parabola has one focus about which the shape is constructed; an ellipse and hyperbola have two. A directrix is a line used to construct and define a conic section. The distance of a directrix from a point on the conic section has a constant ratio to the distance from that point to the focus. As with the focus, a parabola has one directrix, while ellipses and hyperbolas have two.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. Parts of conic sections: The three conic sections with foci and directrices labeled. Each type of conic section is described in greater detail below. A parabola is the set of all points whose distance from a fixed point, called the focus, is equal to the distance from a fixed line, called the directrix. The point halfway between the focus and the directrix is called the vertex of the parabola. In the next figure, four parabolas are graphed as they appear on the coordinate plane. They may open up, down, to the left, or to the right. Four parabolas, opening in various directions: The vertex lies at the midpoint between the directrix and the focus. An ellipse is the set of all points for which the sum of the distances from two fixed points (the foci) is constant. In the case of an ellipse, there are two foci, and two directrices. In the next figure, a typical ellipse is graphed as it appears on the coordinate plane. Ellipse: The sum of the distances from any point on the ellipse to the foci is constant. A hyperbola is the set of all points where the difference between their distances from two fixed points (the foci) is constant. In the case of a hyperbola, there are two foci and two directrices. Hyperbolas also have two asymptotes. A graph of a typical hyperbola appears in the next figure. Hyperbola: The difference of the distances from any point on the ellipse to the foci is constant. The transverse axis is also called the major axis, and the conjugate axis is also called the minor axis. Conic sections are used in many fields of study, particularly to describe shapes. For example, they are used in astronomy to describe the shapes of the orbits of objects in space. Two massive objects in space that interact according to Newton's law of universal gravitation can move in orbits that are in the shape of conic sections. They could follow ellipses, parabolas, or hyperbolas, depending on their properties. Every conic section has a constant eccentricity that provides information about its shape.Discuss how the eccentricity of a conic section describes its behavior
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:
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 . This indicates that the distance between a point on a conic section the nearest directrix is less than the distance between that point and the focus. Conic sections are formed by the intersection of a plane with a cone, and their properties depend on how this intersection occurs.Discuss the properties of different types of conic sections
eee eee . Types of conic sections: This figure shows how the conic sections, in light blue, are the result of a plane intersecting a cone. Image 1 shows a parabola, image 2 shows a circle (bottom) and an ellipse (top), and image 3 shows a hyperbola. A parabola is formed when the plane is parallel to the surface of the cone, resulting in a U-shaped curve that lies on the plane. Every parabola has certain features:
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:
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. Conic sections graphed by eccentricity: This graph shows an ellipse in red, with an example eccentricity value of0.50.50.5 , 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. When the plane's angle relative to the cone is between the outside surface of the cone and the base of the cone, the resulting intersection is an ellipse. The definition of an ellipse includes being parallel to the base of the cone as well, so all circles are a special case of the ellipse. Ellipses have these features:
Ellipses can have a range of eccentricity values: 0≤e<10 \leq e < 10≤e<1 000 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:
(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.CC licensed content, Shared previouslyCC licensed content, Specific attribution |