What is Conic Section?

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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

 Circle (x - h)² +  (y - k)² = r² The coordinates of the center of the circle is (h, k).'r' is the radius of the circle. Hyperbola With Horizontal Transverse Axis (x - h)²/a²- (y - k)²/b² = 1 (h, k) are the coordinates of the center of the hyperbola.Distance between the vertices of the hyperbola is given as 2a whereas the distance between the foci is given as 2c.     c² = a² + b² Hyperbola With Vertical Transverse Axis (y - k)²/a² - (x - h)²/b² = 1 (h,k) are the coordinates of the center of hyperbola.Distance between the vertices of the hyperbola is given as 2a whereas the distance between the foci is given as 2c.     c² = a² + b² Ellipse With Horizontal Axis (x - h)²/a²- (y - k)²/b² = 1 (h,k) are the coordinates of the centre of Ellipse.The length of the major axis is 2aThe length of the minor axis is 2b.Distance between the center and either focus is c with   c² = a² - b², a > b > 0 Ellipse With Vertical Axis (y - k)²/b² - (x - h)²/a² = 1 (h,k) are the coordinates of the center of Ellipse.The length of the major axisx is 2aThe length of the minor axis is 2b.Distance between the center and either focus is c with.   c² = a² - b², a > b > 0 Parabola With Horizontal Axis (y - k)² = 4p(x - h)p ≠ 0 (h,k) are the coordinates of the center of Parabola.The directrix of parabola is defined by the equation x = h - p.Axis is the line y = k. Parabola With Vertical Axis (x - h)² = 4p(y - k)p ≠ 0 (h,k) are the coordinates of the center of Parabola.The directrix of the parabola is defined by the equation x = k - p.Axis is the line x = h

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.

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:

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.

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.

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/- (y - k)²/= 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.

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.

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.

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.

Solved Examples

1. Identify the graph of each equation given below as a parabola, ellipse, circle, or hyperbola.

1. 4a² + 4b² - 1 = 0

2. 3a² - 2b² - 12 = 0

3. a - b² - 6b + 11 = 0

Solution:

1. 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.

1. 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

1. a - b² - 6b + 11 = 0

2. a =  b² + 6b - 11 = 0

3. a = (b² + 6b + 9) + 11 = 9

4. 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.

1. What are the Applications of Conic Section?

Ans: Here are some real-life applications of the conic section:

• The conic section hyperbola is used in a navigation system known as Loran.

• Parabolic mirrors are used to converge light beams at the focus of the parabola.

• Hyperbolic along with the parabolic mirror and lenses are used in the system of telescopes.

• Parabolic mirrors are used in solar cookers to converge light beams to use for heating.

• Conic sections are used in astronomy to describe the shape of the orbit in space.

2. What is the General Form of Conic Section?

Ans: All conic sections are explained by 2nd-degree polynomials in two variables.  The general form of conic section is given as Axy² + Bxy + Cy² + Dx + Ey + F = 0, where A, B, C, and D are constants.

If the coefficient B is zero, then the conic section is formed horizontally or vertically. The equation can be substituted in one of the ways given above by completing the square.

3. Who is Credited with the History of the Conic Section?

Ans: A Greek Mathematician Menaechmus is credited with the history of the conic section.