What is Conics: Definition and 63 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.

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  1. A

    I Canonical Form for quadratic equations *with* linear terms

    Hello: I'm not sure if there's an accepted canonical form for a quadratic equation in two (or more) variables: $$ax^2+by^2+cxy+dx+ey+f=0$$ Is it the following form? (using the orthogonal matrix Q that diagonalizes the quadratic part): $$ w^TDw+[d \ \ e]w+f=0$$ $$w^TDw+Lw+f=0$$ where $$...
  2. I

    How can the vertices of an ellipse be found using polar equations?

    The text says that the following conic, r = 15/[3-2cos(theta)], can be rearranged to 5/[1-(2/3)cos(theta)]. The graph of the conic is an ellipse with e=2/3. Then it says that the vertices lie at (15,0) and (3,pi). How did they find the vertices? Thanks.
  3. José Ricardo

    Conics - How to identify figures in the plane?

    Homework Statement Identify the following figures in the plane, saying if they are circles, ellipses, hyperboles or parabolas and giving their centers, rays, foci, and guidelines as the case may be:Homework Equations x² + y² - 3x = 0 The Attempt at a Solution x² + y² - 3x = 0 x(x - 3) + y² = 0...
  4. Michii

    I Differential equation of all the conics in the plane

    Hi, the problem is parametric families: To find Differential equation of all the conics in the plane with the origin in the center But when you speak of center at the origin being the equation of the conics: Ax ^ 2 + Bxy + cy ^ 2 + Dx + ey + F, is it correct to take the origin by making x and...
  5. S

    A Why are conics indistinguishable in projective geometry?

    It is said that curves of the second order which we usually refer to as ellipse, parabola and hyperbola, i. e. conics, are all represented on projective plane by closed curves (oval curve), which means there is no distinction between them. Why is it? Projective space can, in principle, be...
  6. D

    Downward parabola: value of p = 1/a?

    Homework Statement Hello! I am repeating conics, and have stumbled upon this paragraph on purplemath.com " Then a = –1/9. With a being the leading coefficient from the regular quadratic equation y = ax^2 + bx + c, I also know that the value of 1/a is the same as the value of 4p, so 1/(–1/9) =...
  7. S

    Effects of External Pressure on a Cone

    In my study I deal with tubulars frequently, and it is well known how to calculate stresses due to external pressure on a (hollow) uniformly-thick cylinder (i.e. a pipe). Suppose now that I have a cone, tapering downward like a V, with a hollow cylindrical interior (like the inside of a pipe)...
  8. S

    Tangent to ellipse also tangent to circle

    Homework Statement if the tangent at a point P("theta") on the ellipse 16 (x^2) + 11 (y^2) = 256 is also tangent to the circle (x^2) + (y^2) + 2(x) = 15 then ("theta") = ?? 2. The attempt at a solution {{{{ i have taken "theta" as "d" }}}} P [4 cos d , (16/(sqrt11)) sin d] equation of...
  9. K

    MHB Conics- Word problem with ellipses.

    Hi. I am given the following problem. A small bridge is shaped like a semi-ellipse. Given that its maximum height is 3m and that its foci are located 4m from the centre find the height of the bridge at a distance of 2m from its edge. So the problem give me the values b= 3 and c=4. With this we...
  10. O

    Why Don't My Patched Conics Calculations for Orbital Mechanics Match the Text?

    I'm working on self teaching the patched conics approach for orbital mechanics. I am a very visual learner and I am having difficulty finding text that tailors to my need. Recently I have been working on the calculations for an interplanetary transfer and I seem to be getting pieces of the...
  11. J

    Conics and Beyond: Understanding Trajectories in Multiple Gravitational Fields

    When a object of mass non insignificant is traveling in space and is sob influence of one gravitational field, the trajectory described by this object is a conics (Ax²+Bxy+Cy²+Dx+Ey+F=0), but and if the object is sob influence of two gravitational fields, so the equation the described this...
  12. S

    MHB Find Equation of Ellipse at (1,2) & (1,8), Minor Axis Length 4

    Find equation for center at (1,2) and vertex at (1,8) and Minor axis length of 4?
  13. J

    What is the directrix for a polar conic with the given equation?

    Homework Statement Hi wondering what the directrix is for this Homework Equations r = (20)/ (2+sin(theta)) The Attempt at a Solution I factored denominator so it read 2(1+ (1/2)sin(theta)) ...So I said e times d = (2/20) and I got d = (1/5) because e is (1/2). Doesn't make...
  14. matqkks

    Sketch Conics: Motivating Students to Learn Cross Products

    What is the most motivating way to introduce the sketching of conics which have a cross product terms? This topic involves a lot of other stuff such as eigenvalues, orthogonal matrices, completing the square etc. I find a significant number of students get lost in this forest of sketching...
  15. matqkks

    MHB Sketch Conics: Motivating Students to Learn Cross Product Terms

    What is the most motivating way to introduce the sketching of conics which have a cross product terms? This topic involves a lot of other stuff such as eigenvalues, orthogonal matrices, completing the square etc. I find a significant number of students get lost in this forest of sketching...
  16. matqkks

    MHB Quadratic Forms: Beyond Sketching Conics

    What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?
  17. matqkks

    Quadratic Forms: Beyond Sketching Conics

    What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?
  18. Hysteria X

    Conics Equation and Circle Problem: Solving for Unknown Variables

    Homework Statement The equation ##x^2y^2-2xy^2-3y^2-4x^2y+8xy+12y=0## represents?? Homework Equations circle: ##x^2 +y^2 = a^2## The Attempt at a Solution i know this has something to do with seperating out the variables but i don't seem to get the req equation
  19. S

    MHB What Surprising Patterns Can Everyday Objects Reveal About Conic Sections?

    I made these remarkable "discoveries" while in college. If you have a table lamp with a cylinderical shade, turn it on and look at the pattern on the nearby wall. You will see a hyperbola. I was in the cafeteria, staring into my coffee mug. There was particularly bright source of light nearby...
  20. U

    How Can Two Confocal Conics Pass Through a Single Point?

    Homework Statement Through a given point in the plane of an ellipse prove that exactly two conics (one eliipse and the other hyperbola) confocal with the given ellipse can be drawn. Homework Equations The Attempt at a Solution Let the equation of given ellipse be...
  21. L

    Find Equation of Conic with Vector & Foci - Ellipse

    Homework Statement Given the vector (6,-15) and foci (6,10) (6,-14), Find the equation of the conic.Homework Equations Vector = <6,-15> (x-h)2/a2 + (y-k)2/b2 = 1 Foci (h+c,k) and (h-c,k) vertices (h+a,k), (h-a,k) The Attempt at a Solution k = 6 h=-2 a=-12 I'm not sure what I'm doing. How do...
  22. S

    Please help understanding what the question is asking Calc II (Conics)

    Homework Statement Find an equation is standard form for the conic that satisfies the given condition. Parabola: vertex (2,3), vertical axis passing through (1,5) Homework Equations vertex= (h,k) The Attempt at a Solution I know how to solve a problem like this if I was given the...
  23. T

    Patched Conics Trajectory from Initial Orbit State

    Hi, I've got a problem here, that I'd like to discuss before trying to implement any sort of solution. Basically I've got a planetarium simulation, in which I'd like to plot a patched conic trajectory ahead of a spacecraft . The planetarium only simulates two-body gravity, and switches...
  24. D

    Foci and the definition of a hyperbola

    Homework Statement Find an equation of a hyperbola such that for any point (x,y) on the hyperbola, the difference between its distances from the points (2,2) and (10,2) is 6. Homework Equations -None- The Attempt at a Solution I tried graphing it and making the (10,2) and (2,2)...
  25. B

    Solve Calculus & Conics: Parabolic Reflector, Opening 10cm, Diameter 7cm

    Homework Statement A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is 10 cm. Find an equation of the parabola where V is at the origin. Find the diameter of the opening | CD |, 7 cm from the vertex. Because the...
  26. K

    Calculators Help TI-84 CONICS App is Giving Trig Formulas

    TI-84 HELP! The formulas under the CONICS app keeps converting into trig... It saying a circle is x=rcos(T) + H instead of (x-h)^2 + (y-k)^2 = r^2 How do I get the 'normal' formulas back? PLEASE HELP!
  27. Shackleford

    Finding Tangent Lines on a Hyperbola: Proving Proposition I-34

    This is from my History of Mathematics course. Well, I centered the hyperbola at the origin and found the derivative. It gives me the slope of the tangent line at whatever point I want. How do I prove this?
  28. B

    Real-World Applications of Conic Sections: How Can They Benefit Us?

    So I've been wondering for a long time: How are conics studied in Calculus and Analytic Geometry? Also, how are they applied in real-life situations, whether it's something as simple as construction, or something as complex as physics and engineering. Ever since I studied conics in my...
  29. F

    Why is it wrong to conclude a isn't the vertice (conics)

    Homework Statement Given the ellipse is centered at the origin and the information, find an equation of the ellipse. 1) An ellipse with focus (\sqrt{5},0), directrix; x = \frac{9}{\sqrt{5}} 2) An ellipse with an eccentricity of 4/5 The Attempt at a Solution I have to leave...
  30. F

    Schools Does anyone know which college math course uses conics the most?

    Unfortunately my inferior Canadian education skips out at least a year of math topics in comparison with my American counterparts, so conics weren't a "huge" part of our curriculum. Can someone tell me what course demands existing firm knowledge of properties of parabolas (directix, focus...
  31. E

    How do we prove tangent lines to conics using homogeneous coordinates?

    I am unable to comprehend the proof for tangent line to conics. Here is the proof as per the book (Multiview Geometry by Hartley and Zisserman). Everything is in homogeneous coordinates. The line l = Cx passes through x, since lT x = xT Cx = 0. If l has one-point contact with the conic...
  32. Mentallic

    Solve Conics Hyperbola Homework Statement

    Homework Statement http://img11.imageshack.us/img11/6340/conicshyperbola1.jpg Homework Equations d^2=(x_2-x_1)^2+(y_2-y_1)^2 y-y_1=m(x-x_1) m_1m_2=-1 The Attempt at a Solution I was able to answer (i) but for (ii) I would go about it like this: Find the equation of the line...
  33. R

    Finding Directrix and Sketching Hyperbola in Polar Coordinates

    Homework Statement Hyperbola formula 9x^2 - 4y^2 + 36x + 24y - 36 = 0. Convert to rectangular form, find coordinates of the vertices, find coordinates of the foci, find eccentricity, what is the equation of the conic section in polar coordinates if the pole is taken to be the leftmost focus...
  34. D

    Polar equations of conics question

    for this problem, I've been given the vertices of the hyperbola as (4, pi/2) and (-1, 3pi/2). the question asks to find the polar equation of this hyperbola. so what i did was do a quick sketch of the graph. (4, pi/2) is essentially (0,4) and (-1, 3pi/2) is essentially (0,1). the midpoint of...
  35. C

    What is the Type and Orientation of Conics in Problems a and b?

    Homework Statement There are 2 problems a and b. I've solved for both already. I just need to know how to describe them by type and orientation. In other words, what does what I got tell me in regards to type and orientation and how do I know this for future problems? (e.g. hyperbola or...
  36. X

    Restricting Domain and Range in broken conics

    Homework Statement How do you know whether you restrict the domain or range with a horizontal ellipse ? Homework Equations x^2/49 +y^2/10=1 The Attempt at a Solution
  37. D

    Graphing Conics: Finding the Vertex, Directrix, and Focus of x^2-4y^2+2x+8y-7=0

    Homework Statement graph the following: Homework Equations x^2-4y^2+2x+8y-7=0 The Attempt at a Solution So far I've gotten to (x+1)^2 = -4(y-2) If that's right, I have p = -1 and v = (-1 , 2) and directrix: y=2. Could someone double check me to see if I'm doing it right? Thanks
  38. D

    Conic Sections: Graphing with Multiple Squares

    Homework Statement graph the following Homework Equations 9x^2+4y^2+36x-8y+4=0 The Attempt at a Solution I think I need to get it into \frac{(x-x0)^2}{a^2}+\frac{(y-y0)^2}{b^2} but I'm not sure. I have \frac{9x^2}{-4}-8x+y^2-2y=1 and now I'm stuck
  39. D

    Graph Conics: Center, Verticies, Foci, Asymptote, Directrix

    Homework Statement Graph the following. Include center, verticies, foci, asymptote, and directrix as appropriate. Homework Equations x^2 + 8y - 2x = 7 The Attempt at a Solution So far I have: V = (1, -7/8) P = -2 X = -1 I have no clue where to go from here, or if I'm...
  40. A

    Ploting general equation of conics

    i have made a program that takes input of Ax^2+By^2+Cxy+Dx+Ey+F=0 it determines the type pf curve and finds all the related elements...now i want to ad the GRAPHS as well...but i can't find a way...i just completed my 12th standard so i don't have enough knowlege...even i can't find related...
  41. T

    Conics Problem Part 5: Homework Statement & Equations

    Homework Statement http://img93.imageshack.us/img93/6889/arch4gh8.jpg [/URL] Homework Equations Above The Attempt at a Solution All good?
  42. T

    Solve Conics Problems: Homework Statement for Part 4

    Homework Statement http://img125.imageshack.us/img125/6893/arch5tq8.jpg [/URL] Homework Equations On work The Attempt at a Solution Just need someone to check my work.
  43. T

    Conics Problem Part 3: Equations and Solutions | Homework Statement

    Homework Statement http://img103.imageshack.us/img103/3784/arch3tf0.jpg [/URL] Homework Equations On picture above The Attempt at a Solution Again, I just want someone to check my work.
  44. T

    Conics Problem Part 2: Formulas and Solutions | Homework Statement

    Homework Statement This is the second page. http://img116.imageshack.us/img116/7519/arch2iq9.jpg [/URL] Homework Equations Formulas on picture above. The Attempt at a Solution I'am just wondering if everything looks fine.
  45. E

    Solving Conics: Find the x-Intercept

    Conics! Help Please! okay i think i have solved this correctly...still a little unsure though... y=x^2-6x+3----find x intercept first i used complete the square---- y-3=(x-3)^2 then i solved for x---- 3+(y-3)^(1/2)=x then i plugged 0 in for y and got 3+(3^(1/2)) and 3-(3^(1/2)) is that...
  46. T

    Polar Coordinates and Conics, bad

    Were on the conic section. I need help how to choose the right interval to evaluate the arc lengh. x=5cost-cos5t and y=5sint-sin5t . I don't get how to choose the inverval to evaluate this, can someone pleasse tell me how. I just don't grasp this.
  47. S

    Classification of figure from the general equation of conics

    during my study on conics , I found a formula in the book regarding the classification of figure from the general equation of conics ax2+2hxy+by2+2gx+2fy+c=0 it was given that \Delta=abc+2fgh-af^{2}-{bg}^{2}-{ch}^{2} if \Delta \neq 0 then if h^{2}=ab...parabola h^{2}<ab...ellipse...
  48. K

    Conics - Semi Major/Minor Axis

    Hi, I have this homework question and I completed and found the the foci and the center for the ellipse, but I don't understand how to find the semi major and minor axis. Graph and give the center, semi major and semi minor axis and foci of the ellipse 25x^2 + 350x + 9y^2 - 54y +1081 =...
  49. R

    Determine Bulb Position for Parabolic Reflector of Headlight

    . ...|...B . ...| . ...| . ...| ----|---------D . ...|A . ...| . ...| . ...|...C Remember its a parabola connect the dots. This is what i have done x^2/a^2 - y^2/b^2 = 1 then i substituted a as 0 and b as 16 and it got me nowhere I'm guessing it has to be located at 0 but i...
  50. M

    Solving a Conics Question: Bridge Arch Height at Point 30m from Center

    Hello, I started this problem. I don't know really how to set it up. I attached my work. I know it is wrong but I do not know where. the correct answer is 42.2. Here is the question: A bridge over a river is supported by a hyperbolic arch which is 200 m wide at the base. The maximum...