Conics Project

Conics

Project

Ellipse

Definition

A curve having a constant sum of distances from two fixed points on an XY-plane is known as an ellipse. It is classified as one of the conic sections that are obtained by cutting a cone at an angle with its base. In case the plane intersects the cone parallel to the base, a circle is formed.

Shape

The resulting shape is a closed curve that resembles a stretched circle, with one axis longer than the other, giving it an oval shape.

Major & Minor Axis

The principal axis denotes the most extended diameter of the ellipse, traversing the core from one terminus to the other, at the wider section of the ellipse.

The subordinate axis represents the briefest diameter of the ellipse intersecting through the center at the slimmest section.

Properties

The unchanging length is referred to as a directrix.

An oval has two focus points, which are also known as foci.

An oval has a primary axis, a secondary axis, and a midpoint.

The elongation of an oval ranges from 0 to 1. 0≤e<1

The overall amount of distance from the curve of an oval to the two focus points remains the same.

Eccentricity

The eccentricity of an ellipse is defined as the ratio of the distances from the center of the ellipse to either focus to the semi-major axis of the ellipse. It can be calculated using the formula e = c/a, where c represents the focal length and a represents the length of the semi-major axis. As c is always less than or equal to a, the eccentricity of an ellipse is always less than 1. Additionally, the value of c can be determined using the formula c² = a² - b². By substituting this into the original formula for eccentricity, we can also express it as e = √[(a² - b²)/a²], which can be further simplified to e = √[1 - (b²/a²)].

Equation

The equation of the ellipse is: x2/a2 + y2/b2 = 1

Derivation

The next figure represents an ellipse such that P1F1 + P1F2 = P2F1 + P2F2 = P3F1 + P3F2 is a constant. This constant is always greater than the distance between the two foci. When both the foci are joined with the help of a line segment then the mid-point of this line segment joining the foci is known as the center, O represents the center of the ellipse in the figure given below:

The line segment passing through the foci of the ellipse is the major axis and the line segment perpendicular to the major axis and passing through the center of the ellipse is the minor axis. The end points A and B as shown are known as the vertices which represent the intersection of major axes with the ellipse. ‘2a’ denotes the length of the major axis and ‘a’ is the length of the semi-major axis. ‘2b’ is the length of the minor axis and ‘b’ is the length of the semi-minor axis. ‘2c’ represents the distance between two foci.

Standard Equation

The typical formula for an ellipse is x2/a2 + y2/b2 = 1. This formula establishes an ellipse situated at the center. In case a is greater than b, the ellipse is elongated more along the horizontal axis, and if b is greater than a, the ellipse is elongated more along the vertical axis.

Formula

Area of Ellipse

The formula for the area of an ellipse is πab, where a and b represent the size of the ellipse's semi-major and semi-minor axes. Like the parabola and hyperbola, both of which are open-shaped and unbounded, the ellipse belongs to the conic section family.

Perimeter

Latus Rectum

The latus rectum is defined as the line segments that are perpendicular to the major axis and pass through any of the foci, while their endpoints lie on the ellipse. The length of the latus rectum can be calculated using the formula L = 2b2/a, where a and b represent the length of the major and minor axis respectively.

+ = 1

(x - 0)2 (y - 1)2 1 4

+ = 1

(x - h)2 (y - k)2 a2 b2

x2 + = 1

(y - 1)2 4

Right Most Point : (1, 1) Left Most Point : (-1, 1) Top Most Point : (0, 3) Bottom Most Point : (0, -1)

Left Most Point : (-1, 1) Bottom Most Point : (0, -1)

h=0 k=1 a=1 b=2

(0, 3) (0, -1)

(-1, 1) (1, 1)

Problem Example

Real Life Examples

Planets travel around the Sun in elliptical routes at one focus.

Mirrors used to direct light beams at the focus of the parabola are parabolic.

Hyperbola

Definition

Its two curves that are like infinite bows. It is the result of a double right intersection on a conic.

The hyperbola has 2 asymptotes, they are two straight lines that pass through the upper part of one bow and the lower part of the other bow.

|PF − PG| = constant

Distance

The distance from P to F is always less than the distance P to G by some constant amount.

Asymptote equations: y= (b/a)x y=- (b/a)x

• PF is the distance P to F
• PG is the distance P to G
• || is the absolute value function

Every hyperbola has a directrix which is a straight line that has the same ratio, this ratio is called eccentricity.

Each bow is called branch, F and G are called focus, they cannot move because they are fixed.

Eccentricity Formula

Latus Rectum

Is a line that passes through the focus and is parallel to the directrix

Fórmula: 2b2/a 1/x Is the reciprocal of a hyperbola

- = 1

(y - k)2 (x - h)2 b2 a2

h=2 k=0 a=3 b=4

y = 0 + (x - 2) = x -

4 4 8 3 3 3

y = 0 - (x - 2) = - x +

4 4 8 3 3 3

Problem Example

Real Life Examples

A guitar is an example of hyperbola as its sides form hyperbola.

The Kobe Port Tower has hourglass shape, that means it has two hyperbolas.

APA References

Admin. “Ellipse (Definition, Equation, Properties, Eccentricity, Formulas).” BYJUS, 4 Apr. 2023, byjus.com/maths/ellipse/#definition.

“Lesson 41: Conic Sections: Ellipses.” Lesson 41: Conic Sections: Ellipses | Mathematical Association of America, www.maa.org/programs/faculty-and-departments/course-communities/lesson-41-conic-sections-ellipses. Accessed 10 May 2023.

https://math.libretexts.org/Bookshelves/Algebra/Map%3A_College_Algebra_(OpenStax)/08%3A_Analytic_Geometry/8.02%3A_The_Ellipse#:~:text=Thus%2C%20the%20standard%20equation%20of,further%20in%20the%20 vertical%20direction.

“Applications of Conic Sections3.” Share and Discover Knowledge on SlideShare, www.slideshare.net/iramkhan66/applications-of-conic-sections3#:~:text=Bicycle%20chain%20is%20an%20example%20of%20ellipse.&text=Earth%20orbit%20around%20the%20sun,Without%20the%20orbit%20we%20all%20. Accessed 12 May 2023.

“12.1: The Ellipse.” Mathematics LibreTexts, 17 Dec. 2020, math.libretexts.org/Courses/Prince_Georges_Community_College/MAT_1350%3A_Precalculus_Part_I/12%3A_Analytic_Geometry/12.01%3A_The_Ellipse#:~:text=Many%20real%2d World%20 situations%20can,stones%20by%20 generating%20sound%20 waves.

“Hyperbola.” Math Is Fun, www.mathsisfun.com/geometry/hyperbola.html. Accessed 16 May 2023.

“Hyperbola - Equation, Properties, Examples: Hyperbola Formula.” Cuemath, www.cuemath.com/geometry/hyperbola/. Accessed 16 May 2023.

“Applications of Conics in Real Life: Conic Sections.” Cuemath, www.cuemath.com/learn/mathematics/conics-in-real-life/#P005. Accessed 16 May 2023.

Thank You

Luisa Sánchez Villarreal