What is Conic sections: Definition and 37 Discussions

In mathematics, 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 of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.
The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, and some particular line, called a directrix, are in a fixed ratio, called the eccentricity. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables, which may be written in matrix form. This equation allows deducing and expressing algebraically the geometric properties of conic sections.
In the Euclidean plane, the three types of conic sections appear quite different, but share many properties. By extending the Euclidean plane to include a line at infinity, obtaining a projective plane, the apparent difference vanishes: the branches of a hyperbola meet in two points at infinity, making it a single closed curve; and the two ends of a parabola meet to make it a closed curve tangent to the line at infinity. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically.

View More On Wikipedia.org
  1. A

    Equation of hyperbola confocal with ellipse having same principal axes

    The equation of ellipse reduces to : $$(2x+3)^2+(3y+2)^2=8$$ $$\frac{(x+3/2)^2}{8/4}+\frac{(y+2/3)^2}{8/9}=1$$ Center of ellipse =##\left(\frac{-3}{2},\frac{-2}{3}\right)## ##b^2=a^2(1-e^2)=8/9## and ##a^2=8/4## Therefore ##e=\frac{\sqrt{5}}{3}## Distance between foci=##\frac{2\sqrt{10}}{3}##...
  2. A

    I Converse of focus-directrix property of conic sections

    In my recent study of Conic Sections, I have come across several proofs (many of those comprise Dandelin spheres) showing that the cross-section of a cone indeed follows the focus-directrix property: "For a section of a cone, the distance from a fixed point (the focus) is proportional to the...
  3. theycallmevirgo

    Simple(?) parabolic dish design question

    I want to use this to design a parabolic (optical) mirror; The problem is that in my application I need both D and f to be a parameter, but I need to specify f only as a perpendicular distance from D. In other words, I need to specify some f_2=f-d, and calculate d. I can't seem to come up with...
  4. G

    Different types of hyperbolas and their properties

    I know the hyperbola of the form x^2/a^2-y^2/b^2=1 and xy=c; but coming across this question I'm put in a dilemma of how to proceed with calculating anything of it - say eccentricity or latus rectum or transverse axis as said. How to generalize a hyperbola (but i don't want a complex derivation...
  5. JTC

    I Hypatia, Elliptic orbits, conic sections

    I understand that Hypatia proposed elliptic orbits. I also understand she studied conic sections. And, of course, one of the intersections of an inclined plane through a cone is elliptical. So now my question: if she did propose elliptic orbits (not interested in who may also have done it)...
  6. Adgorn

    I Spivak's Calculus: clarification on Conic Sections appendix

    Hello everyone. This was originally a homework problem but I realized my misunderstanding stems from the explanation given before the problem so here we are. The thread deals with these 3 pages from Spivak's Calculus: https://ibb.co/kAKyVU https://ibb.co/jXVSPp https://ibb.co/kwRdVU I'm pretty...
  7. Adgorn

    Spivak: Conic Sections appendix, problem 1

    Homework Statement "Consider a cylinder with a generator perpendicular to the horizontal plane; the only requirement for a point ##(x,y,z)## to lie on this cylinder is that ##(x,y## lies on a circle: ##x^2+y^2=C^2##. Show that the intersection of a plane with this cylinder can be described by...
  8. M

    Y=cx[L−x], The value of the constant c for a perfect circle

    Homework Statement For the equation y=cx[L−x] say for a circle with the value of L at 100 meters and the value of x at 25 meters. What would be the value of the constant c for a perfect circle. 3. Attempt at the Solution: I can approximate and graph this with different values of c however I'm...
  9. A

    Points on an elliptical orbit where V(radia) is zero.

    Homework Statement Points on an elliptical orbit where the speed is equal to that on a circular orbit? Homework EquationsThe Attempt at a Solution I have attempted this question and my calculations show that at points on minor and major axes, the radial component of velocity is zero. Hence at...
  10. S

    I Question inspired by reviewing conic sections

    Recently, in my calculus two class, we began going over conic sections. After reviewing the definitions of ellipses and hyperbolas - For two given points, the foci, an ellipse is the locus of points such that the sum of/difference between the distance to each focus is constant, respectively - I...
  11. S

    Trajectories of planets using reduced mass and CM frame

    In planetary motion, the reduced mass of a system \mu is used in order to study the motion of the planet m in the non-inertial frame of the star M. Using \mu the trajectory of m turns out to be a conic. But this is the trajectory of the planet m as seen from the star M, correct? I read that in...
  12. Erika Brown

    Wind Turbines and Conic Sections

    I am working on a project for my precalculus class. They are going to study wind turbines and design a wind turbine blade. I am trying to understand the math behind wind turbines myself. Can I relate wind turbines to a study of conic sections?
  13. S

    Geometry Conic Sections: Finding the Perfect Book

    I need a good book with good problems and teaching for conic sections.
  14. B

    Drawing Curves with Conic Sections

    I'd like of draw any curve using combination of line circle, elipse, parabola, hyperbola and straight. Of course several curves can't be designed with 100% of precision using just conic curves, but, can to be approximated. Acttualy, I don't want to reproduce a curve already designed but yes...
  15. Greg Bernhardt

    Understanding Conic Sections: Exploring the Plane-Cone Intersection

    Definition/Summary A conic section (or conic) is any curve which results from a plane slicing through an upright circular cone. If the slope of the plane is zero, it cuts only one half of the cone, and the conic is a circle (or a point, if the plane goes through the apex of the cone). If the...
  16. S

    Finding Focus Points of Parabolas in Conic Sections

    Hi all, I am having an issue with the following problem. I just don't know how to approach it. Homework Statement Homework Equations Ax^2 + Bxy + Cy2 + Dx + Ey + F = 0 The Attempt at a Solution I am confused on how to put this problem in terms of x & y and get numerical values for both...
  17. L

    Conic sections vs multivariable functions

    Hello, We just started to learn about functions of several variables in my Calculus class and my question is simple: Are conic sections, like ellipses, multivariable functions or is y still dependant on x? Are ellipses just single variable functions slightly rearranged? Thanks in advance...
  18. Ackbach

    MHB Exploring the Relationship Between Conic Sections & Orbits

    If you think about a double-napped cone, and the various non-degenerate sections you can get with it: 1. Circle 2. Ellipse 3. Parabola 4. Hyperbola, you can see that there is a progression here: increasing angle $\alpha$ that the intersecting plane makes with the horizontal. To be clear about...
  19. 3

    Exploring Conic Sections in Everyday Objects: The Case of a Cup of Tea

    Homework Statement I was sitting drinking a cup of tea earlier. The cup was of course cylindrical. I was just gazing into the cup looking at the tea as my cup was flat and the top level of the tea looked like a circle. Watching the tea as I tilted the cup to drink I noticed that the shape...
  20. MarkFL

    MHB Proof and tutorial on conic sections

    Conic sections are the curves that are formed when a plane intersects the surface of a right circular cone. These curves are the circle, ellipse, hyperbola, and the parabola. The study of conic sections dates back over 2000 years to ancient Greece. Apollonius of Perga (262-190 B.C.) wrote an...
  21. B

    Conic sections: Can focal points be outside the ellipse?

    Can an ellipse's focal points be outside the ellipse? I have tried googling this, but without any good explanations or answers. According to my calculations, the focal points of the ellipse defined by x^{2} + \frac{y^{2}}{4} = 1 are (-\sqrt{3},0) (\sqrt{3},0)) . I maybe wrong of course...
  22. M

    Understanding Elliptical Orbits: Foci, Center, Barycenter, and Mass Relationship

    What is the relationship between the following in an elliptical orbit: -foci -geometric center -barycenter -the more massive body/the less massive body Also, what path does the larger mass take?
  23. M

    Conic Sections on the Complex Plane (circle)

    Homework Statement Describe the locus and determine the Cartesian Equation of: \left|z-3-5i\right|= 2 Homework Equations \left|z-C\right|= r -----> formula for a circle on complex plane Where C = the centre z = the moving point (locus) (x-h)^{2}+(y-k)^{2}=r^{2} -----> Formula...
  24. Q

    Exploring Conic Sections: What are They and Why are They Important?

    I'm totally confused...i don't understand conic sections. Why do these curves so special? They are described as results of a cone cut by a plane and the equations are constructed from the distances from a point and directrix. We find evidence of these curves everywhere-in projectiles, in...
  25. K

    Teaching Conic Sections: Tips & Strategies

    So I currently teach a precalc class and new this year we are required to teach conic section. We cover parabolas, circles, ellipses, and hyperbolas. Since I haven't taught this before, I was wondering if anyone has suggestions on how to teach it? The book we use has a bunch of formulas, but...
  26. V

    Your thoughts on conic sections

    I need some feedback about something that does not make sense. The parabola and hyberbola can be found in the conic sections. These curves are seen if one looks at right angles to the plane of the section (cut surface if you will).. All math books also say that the ellipse is a result of a...
  27. M

    What Determines the Shape of a Quadratic Equation's Graph?

    the set of points described by the quadratic equation a y^2 + b xy + c x^2 + d y + e x + f = 0 can be 1) a parabola, an ellipse, an hyperbola or 2) an empty set, a line, two intersecting lines, two parallel lines, a circle, a point, and pherhaps something else... I want two know which...
  28. S

    Lens shapes other than conic sections

    This is just a curiosity question. Are there lens shapes different than just conic sections? Any research or any found applications?
  29. C

    Deriving Descriptions of Conic Sections from Fundamental Definition

    Everyone knows by now that a conic section is the figure formed when a plane intersects a right circular cone. Most everyone also knows that there are many different ways to describe a conic, geometrically and algebraically. What one seldom sees is the derivation of those descriptions from the...
  30. G

    Questions about conic sections

    Given the equation of a conic section, how can I: 1) find its focii 2) find the equations of its directrices 3) find out what type of conic it is, without using either the arduous matrix method or the equally arduous rotation method To be honest, I don't really like conic sections...
  31. M

    Some problems with conic sections

    Here are some conic questions that am having problems with : 1) The general form of a particular ellipse is show Below. If the conic is translated 2 units left and 1 unit down, determine the new general term. 2x2+y2-2x+3y-9=0 2) Find the equation of a hyperbola in standard form or...
  32. N

    Conic sections - quadratic curve

    Here is what I know: 1) All quadratic curves of 2 variables correspond to a conic section. ax^2 + 2bxy +cy^2 + 2dx + 2fy + g = 0 a, b, c are not all 0 2) The definitions of parabola (in terms of a directrix and focus), ellipse (in terms of 2 foci), hyperbola (in terms of directrix and...
  33. G

    Conic sections in polar coordinates

    [SOLVED] conic sections in polar coordinates Homework Statement write a polar equation of a conic with the focus at the origin and the given data. i know it's an ellipse with eccentricity 0.8 and vertex (1, pie/2) The Attempt at a Solution my question is: how do I find the...
  34. T

    How Do You Find the Equation of an Ellipse Given Its Center, Vertex, and Focus?

    Homework Statement Center is at (4, -1) Vertex is at (4, -5) Focus is at (4, -3.5) Find the equation of the ellipse. Homework Equations horizontal ellipse: ((x-h)^2)/(a^2)) + ((y-k)^2)/(b^2)) = 1 Vertical ellipse: ((y-k)^2)/(a^2)) + ((x-h)^2)/(b^2)) = 1 c^2 = a^2 - b^2 The...
  35. D

    What is the Definition of a Conic in Complex Projective 2 Space?

    I was just wondering what the more fundamental definition of a conic in complex projective 2 space is. Is it better to say that it is a curve such that the dehomogenisation of its defining equation is a represents a conic in R^2; OR simply a curve defined by a homogeneous degree two polynomial...
  36. P

    Understanding Conic Sections: Ellipse, Parabola, and Hyperbola Explained

    Here's the question: Consider the equation: Ax^2+Cy^2+Dx+Ey+F=0 Consider the cases AC>0, AC=0 and AC<0 and show that they lead to an ellipse, parabola and hyperbola respectively, except for certain degenerate cases. Discuss these degenerate cases and the curves that arise from them Don't...
  37. A

    Conic sections: what can I expect?

    We are doing conic sections, and the practice problems are pretty easy, far too east to be on on of our tests. Can someone give me an example of what I might be asked to find and from what information in a difficult conics question?