MATH CORE TERM 2 PROJECT Done by: Mohamed Saeed AlSayyah & Abdullah Aljasmi and Ahmed Salem 12-4

Introduction Traditionally, the three types of conic section are the hyperbola the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix definition of a conic the circle is a limiting case with eccentricity 0. In modern geometry certain degenerate cases, such as the union of two lines, are included as conics as well.

HISTORY Conic sections are among the oldest curves, and is a oldest math subject studied systematically and thoroughly. The conics seems to have been discovered by Menaechmus (a Greek, ≈375 BCE 〜 325 BCE), tutor to Alexander the Great. They were conceived in a attempt to solve the three famous problems of trisecting the angle, duplicating the cube, and squaring the circle. The conics were first defined as the intersection of a right circular cone of varying vertex angle; a plane perpendicular to a element of the cone. An element of a cone is any line that makes up the cone Depending the angle is less than, equal to, or greater than 90 degrees, we get ellipse, parabola, or hyperbola respectively. Appollonius (≈262 BCE 〜 190 BCE) (known as The Great Geometer) consolidated and extended previous results of conics into a monograph Conic Sections, consisting of eight books with 487 propositions. Appollonius’ Conic Sections and Euclid's Elements may represent the quintessence of Greek mathematics.

Task 1 – Types of conics CIRCLE Each of the geometric figures is obtained by intersecting a double- napped right circular cone with a plane. Thus, the figures are called conic sections or conics. If the plane cuts completely across one nappe of the cone and is perpendicular to the axis of the cone, the curve of the section is called a circle.

ELLIPSE If the plane isn't perpendicular to the axis of the cone, it is called an ellipse. An ellipse is the set of all points in a plane; the sum of the distances from two fixed points in the plane is constant. Many comets have elliptical orbits.

PARABOLA If the plane doesn't cut across one entire nappe or intersect both nappes, the curve of the intersection is called a parabola. A parabola is the set of all points in a plane equidistant from a fixed point and a fixed line in the plane.

HYPERBOLA If the plane cuts through both nappes of the cone, the curve is called a hyperbola. The hyperbola is the set of all points in a plane. The difference of whose distance from two fixed points in the plane is the positive constant.

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Task 2- Comprehensive comparison between conics

Other properties- Ellipse

Parabola

Hyperbola

Classifications

Task 3- Question 1 You can find the equation of a line by knowing two points from that line, know to find and equation of parabola you need to know three points. Find the equation of a parabola that pass through (0,3), (-2, 7) and (1, 4). Parabola

Question 2 - Circle If you have a line equation x+2y=2 and circle equation x 2 +y 2 =25. How many points the graphs of these two equations have in common.

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Task 4- Physics The path of any thrown ball is parabola. Suppose a ball is thrown from ground level, reach a maximum height of 20 meters of and hits the ground 80 meters from where it was thrown. Find the equation of the parabolic path of the ball, assume the focus is on the ground level.

It takes about 76 years to orbit the Sun, and since it’s path is an ellipse so we can say that its movements is periodic. But many other comets travel in paths that resemble hyperbolas and we see it only once. Now if a comet follows a path that is one branch of a hyperbola. Suppose the comet is 30 million miles farther from the Sun than from the Earth. Determine the equation of the hyperbola centered at the origin for the path of the comet. Question 2- Halley’s Comet

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