Conic Section
In Geometry, the conic section, also known as conic, is a curve that is formed by the intersection of a plane and a right circular cone. Conic sections are classified into four groups namely Circle, Parabola, Hyperbola, and Ellipses. None of the conic sections will pass through the vertices of the cone. Conic sections received their name because each conic section is represented by a conic section of a plane cutting through cones.
Conic sections are widely used in Physics, Optical Mechanics, orbits, and others. If the right-circular cone is formed by the plane perpendicular to the axis of the cone, the intersection is considered a Circle. If the plane intersects on one of the pieces of the cone and its axis and not perpendicular to the axis, the intersection will be an ellipse. To form a parabola, the intersection plane must be parallel to one side of the cone and it should intersect one piece of the cone. And, at last, to form a hyperbola, the plane intersects both pieces of the cone. In this case, the slope of the intersecting plane should be more than that of the cone.
Conic Section Definition
A conic section is defined as a curve obtained as the intersection of the cone with a plane. Hyperbola, Parabola, and Circle are three types of conic sections. The circle is a special case of the ellipse and often considered as the fourth type of conic section.
Conic Equation
The general conic equation for any of the conic section is given by:
Axy² + Bxy + Cy² + Dx + Ey + F = 0
Where A, B, C, and D are constants. The shape of the corresponding conic gets changed as the value of the constant changes. If the constant B is zero, then the conic section is formed either horizontally or vertically.
The standard form of conic section equation for each of the conic section is given below:
Standard Form of Conic Section Equations
Graphing Conic Sections
A conic section is a curve formed from the intersection of the right circular cone and a plane. The curves of the conic sections are best explained with the use of a plane and two napped cones. Conic sections are formed when a plane intersects the two napped cones. The graphing conic sections show how a plane and two napped cones form parabola, circle, ellipse, and hyperbola. Let us now understand the graphs of different conic sections.
(Image will be uploaded soon)
Circle
A circle is defined in terms of points, known as the center, and a non-zero length known as the radius. A circle is a locus of points located a radius away from the center. The conic section equation of a circle is (x - h)² + (y - k)² = r². Here, (h, k) are the coordinates of the center and are the radius of the circle.
The graph of a conic Circle as per its equation is given below:
(Image will be uploaded soon)
Parabola
A parabola is defined in terms of line, known as directrix, and the point not on line is known as the focus. A parabola is the locus of points that are equidistant from both the focus and directrix. The axis of symmetry is the line that divides the parabola symmetrically whereas the vertex of the parabola is the intersection of the parabola and axis of symmetry.
The equation of the parabola which opens horizontally is (y - k)² = 4p(x - h), p ≠ 0.
Here (h, k) are the coordinates of the vertex. The directrix according to the equation is given as x = h - p. The focus of the parabola has coordinates (h + p, k). The parabola graph shown below shows how horizontal parabola looks in terms of its equation.
(Image will be uploaded soon)
The equation of the parabola which opens vertically is (x - h)² = 4p(y - k), p ≠ 0.
Here (h, k) are the coordinates of the vertex. The directrix according to the equation is given as y = k - p. The focus of the parabola has coordinates (h, k + p). The parabola graph shown below shows how vertical parabola looks in terms of its equation.
(Image will be uploaded soon)
Ellipse
An ellipse is defined in terms of two points known as foci. An ellipse is the locus of points for which the sum of the distance to each focus is constant. The constant amount is equivalent to the length of the major axis. The general equation of the ellipse is given as (x - h)²/a² - (y - k)²/b² = 1. Here (h, k) are the coordinates of the center of the ellipse. The center of the ellipse is the midpoint of two foci. The chord which passes through two foci is known as the major axis whereas the chord that passes through the center and is perpendicular to the major axis is known as the minor axis.
If a > b, then the ellipse will have a horizontal major axis of length 2a and a vertical minor axis of length 2b.
The foci of the ellipse is located at \[(h - \sqrt{a^{2} - b^{2}},k\], and \[(h + \sqrt{a^{2} - b^{2}},k\]. The graph of an ellipse when a > b is given below.
(Image will be uploaded soon)
If a < b, then the ellipse will have a vertical major axis of length 2b and a vertical minor axis of length 2a.
The foci of the ellipse is located at \[(h, k - \sqrt{b^{2} - a^{2}}\], and \[(h, k + \sqrt{b^{2} - a^{2}}\]. The graph of an ellipse when a < b is given below.
(Image will be uploaded soon)
If a = b, then the ellipse is considered as a circle.
Hyperbola
A hyperbola is a set of all points (x, y) such that the difference of the distances between (x, y) and two different points is constant. The fixed point of the foci is known as a hyperbola. The hyperbola graph has two parts known as branches. Each part looks like a parabola, but slightly different in shape. A hyperbola has two vertices that lie on the axis of symmetry known as the transverse axis. The transverse axis of the hyperbola can be either horizontal or vertical.
(Image will be uploaded soon)
To graph a hyperbola, centered at the origin, first draw a reference rectangle. A rectangle can be drawn with the help of the points (a, b), (-a, b), (a, -b), (-a, -b). Asymptotes of hyperbola lie on the diagonals of the rectangle. The branches of the hyperbola are constructed to approach the asymptotes. The graph of the hyperbola with center at origin is shown below.
(Image will be uploaded soon)
Solved Examples
1. Identify the graph of each equation given below as a parabola, ellipse, circle, or hyperbola.
4a² + 4b² - 1 = 0
3a² - 2b² - 12 = 0
a - b² - 6b + 11 = 0
Solution:
The equation 4a² + 4b² - 1 = 0 is quadratic in both a and b where the leading coefficient for both the variables is the same i.e. 4.
4a² + 4b² - 1 = 0
4a² + 4b² = 1
a² + b² = 1/4
Therefore, equation (a) is the equation of a circle at the origin with a radius of ½.
The equation 3a² - 2b² - 12 = 0 is quadratic in both a and b where the leading coefficient for both the variables has different signs.
3a² - 2b² - 12 = 0
(3a - 2b²)/12 = 12/12
x²/4 - y²/6 = 1
Therefore, equation (b) is the equation of hyperbola opening left and right centered at the origin.
2. The equation a - b² - 6b + 11 = 0 is quadratic in b only
a - b² - 6b + 11 = 0
a = b² + 6b - 11 = 0
a = (b² + 6b + 9) + 11 = 9
a = (b - 3)² + 2
Therefore, equation (c) is the equation of parabola opening right with vertices (2, 3).
3. Find the equation of a circle with radius r and radius centre at (a,0).
Solution:
Given, Centre = (h, k) = (0, 0), and Radius = r
Therefore, the equation of the circle is x² + y² = r².
Facts to Remember
Conic sections are obtained by intersecting a plane with a cone. A cone has two equivalent shaped parts known as nappes. The shape of one nappe looks like a party hat.
An Ancient Greek Geometer and Astronomer Apollonius of Perga is known for his work on the conic section.
FAQs on What is Conic Section?
1. What is a conic section in mathematics?
A conic section is a two-dimensional curve formed by the intersection of a plane and a double-napped right circular cone. The specific type of curve—be it a circle, ellipse, parabola, or hyperbola—is determined by the angle of the intersecting plane relative to the cone's axis and surface.
2. What are the four main types of conic sections?
The four fundamental types of conic sections are derived from how a plane slices a cone. They are:
- Circle: Formed when the plane intersects the cone perpendicular to its axis.
- Ellipse: Created when the plane is tilted but cuts completely through one nappe of the cone.
- Parabola: Results when the intersecting plane is parallel to a generator line of the cone.
- Hyperbola: Formed when the plane is steep enough to intersect both nappes of the cone.
3. How can you identify the type of conic section from its general equation?
You can identify the type of conic from its general second-degree equation, Ax² + Bxy + Cy² + Dx + Ey + F = 0, by calculating the discriminant, which is B² - 4AC.
- If B² - 4AC < 0, the conic is an ellipse or a circle.
- If B² - 4AC = 0, the conic is a parabola.
- If B² - 4AC > 0, the conic is a hyperbola.
4. What are some real-world examples and applications of conic sections?
Conic sections appear frequently in science, engineering, and nature. Key examples include:
- Parabolas: The design of satellite dishes, car headlights, and suspension bridges. The path of a projectile under gravity is also parabolic.
- Ellipses: The orbital paths of planets, moons, and comets around a star. They are also used in medical equipment like lithotripters.
- Hyperbolas: Used in navigation systems (LORAN), the shape of cooling towers, and to describe the trajectories of objects escaping a gravitational field.
5. Are conic sections considered 2D or 3D figures?
This is a common point of confusion. While a conic section is generated by slicing a 3D object (a cone) with a plane, the resulting curve itself lies entirely within that 2D plane. Therefore, a circle, parabola, ellipse, or hyperbola is fundamentally a two-dimensional (2D) figure.
6. Why is the study of conic sections important in fields like astronomy and physics?
The study of conic sections is crucial because these curves mathematically describe motion under the influence of gravity. Johannes Kepler discovered that planets travel in elliptical orbits with the Sun at one focus. Similarly, the paths of comets and other celestial bodies can be modelled as ellipses, parabolas, or hyperbolas. In physics, parabolic surfaces are essential for focusing energy in telescopes and antennas, demonstrating the practical power of these geometric shapes.
7. What is the difference between a circle and an ellipse as conic sections?
A circle is actually a special case of an ellipse. The key difference lies in their eccentricity (e) and focal points. An ellipse has two foci and an eccentricity between 0 and 1 (0 < e < 1). A circle has only one center (both foci coincide), and its eccentricity is exactly 0. Geometrically, a circle is formed by a plane cutting the cone perpendicular to its axis, while an ellipse is formed by a plane cutting at a slight angle.
