The Ellipse Conics Conics are curves obtained by

The Ellipse

Conics • Conics are curves obtained by cutting a double cone with a plane. • Ellipse is a oval closed curve obtained by cutting only one half of a double cone. In particular, if the plane is perpendicular to the axis of the cone, we get a circle.

• Conics were studied by mathematician Apollonius of Perga (262 BC - 190 BC) who moved to Alexandria to explore their properties under the guidance of the great Euclid. • Apollonius, in his work Conic Sections (230 BC), collected 487 theorems relating to conical and remained the most famous treatise on the subject.

The work of Apollonius was commented by the first woman mathematician in history: Hypatia (370 -415). Daughter of the mathematician Theon of Alexandria, Hypatia in 400 was the head of the Platonic school of Alexandria, where she lectured in philosophy and mathematics. It is not known if she has made valuable contributions to the development of mathematics, but she helped her father in the drafting of a commentary of Ptolemy's Almagest (100 -175). It is also thought that Hypatia has brought the first law of Kepler (1561 - 1630) forward according to which planets describe elliptical orbits around the sun which occupies one of the foci. Among the students of Hypatia there were several representatives of Christianity which was then expanding: among them Synesius, who wrote to his mistress some letters in which he praised Hypathia for her teaching ability. Because of her great oratorical arts, Hypatia was considered an obstacle to the development of Christianity; it was thought that her philosophy sank roots in paganism. When in 412 Cyril, new Patriarch of Alexandria, clashed with the Roman prefect Orestes, who was a friend of Hypatia, the woman became a target audience, was killed and cut to pieces by the crowd armed with sharp stones.

Conic sections were studied later by Descartes (1596 - 1650) in his treatise "Geometry" (1637), in which he formulated the methods of analytic geometry.

Ellipse equation • Ellipse is the locus of points, for which a sum of distances from two given points F 1 and F 2 , called first focus and second focus of ellipse, is a constant value.

: focal distance a is the semi-major axis b is the semi-minor axis 2 a > 2 c → a > c

If the foci are on the x-axis: F 1(-c; 0) F 2(c; 0) the equation of ellipse is This is called canonical* equation of the ellipse. a 2 – c 2 = b 2 *canonical from the greek “kanon” which means “norm”

Exercise: Verify that the equation 4 x 2 +9 y 2 = 16 represents an ellipse. Solution: First of all it is necessary to reduce the equation to a canonical one. This means that we have to divide each term of the equation by 16 so as to obtain 1 to the right side of the equation. This is the equation of an ellipse with a 2 = 4 and b 2 = 16/9 a = 2 and b = 4/3. Note: The circle can be considered a particular ellipse with a=b. In this case we get: x 2 + y 2 = a 2 which represents a circle with the center in the origin of axes and with radius equals a. In fact, the foci of the ellipse coincide with the center of the circle, because it’s c = 0.

Draw an ellipse To draw an ellipse just find the intersepts of the curve with the coordinate axes. The points A, A’, B, B’ are the so-called “vertices of the ellipse”

Coordinates of the foci of an ellipse Given the canonical equation of an ellipse, it is possible to determine the coordinates of the foci. Starting from the equality a 2 – c 2 = b 2 we obtain :

Exercise: a = 2 b = 1 c = √(4 -1)=√ 3 The foci are: F 1 (-√ 3; 0) F 2 (√ 3; 0)

Ellipse with the foci on the y-axis Consider an ellipse with the foci on the y-axis and the center in O(0; 0). The coordinates of the foci then will be: F 1(0; - c) and F 2(0; c). Called 2 b the length of the semi-major axis ( this means the axis which contains the foci), the ellipse is defined by the equality: The equation that describes the ellipse is still the same: But, in this case, we have:

Exercise page 405 n° 167 Write the equation of the locus of points, for which the sum of distances from the points (- 2; 0) and (2; 0) equals 14. The points represent the foci of the curve and from their coordinates we can deduce that they are on the x-axis. Hence, c = 2, 2 a = 12 and: The required equation is therefore:

Exercise page 405 n° 168 Write the equation of the locus of points, for which the sum of distances from the points A(0; - 1) and B(0; 1) equals 12. The points A and B represent the foci of the curve and from their coordinates we can deduce that they are on the y-axis. Hence, c = 1, 2 b = 12 and: The required equation is therefore:

Symmetries in the ellipse For each point of an ellipse P, there exist its symmetrical P 2, P 3, P 4 relative respectively to y-axis, to the origin and to x-axis. The x-axis and y-axis are axis of simmetry for the ellipse. The origin is center of simmetry for the ellipse.

Eccentricity The ratio of the focal distance to the length of the major axis of an ellipse is said eccentricity e. 2 c : focal distance 2 a : length of major axis e = 2 c/2 a = c/a = √(a 2 – b 2)/a Since c < a, it follows : 0 < e < 1 The eccentricity of an ellipse is a measure of how nearly circular the ellipse is.

A circle is a special case of an ellipse. Similarly to the fact that a square is a kind of rectangle, a circle is a special case of an ellipse. In terms of the eccentricity, a circle is an ellipse in which the eccentricity equals zero.

Exercise Use the formula to determine 1. the equation of the ellipse 2. the eccentricity of the ellipse below.

a=3 b=2 C=√(9 -4)=√ 5 e = c/a = √ 5/3 Solution

Which among the following equations represent ellipses? If so, write their canonical equation, find the measure of the axis, the vertex and foci coordinates, the eccentricity, and sketch the graph.