Definition: A conic section is the intersection of a plane and a cone.

The conics

History   Conic sections are among the oldest curves studied,   Is the oldest math subject studied systematically and thoroughly.   The conics seems to have been discovered by Menaechmus (a Greek, c BC), tutor to Alexander the Great.   They were conceived in an attempt to solve the three famous problems of trisecting the angle, duplicating the cube, and squaring the circle.

Geometric Problems of Antiquity   The Greek problems of antiquity were a set of geometric problems whose solution was sought using only compass and straightedge: 1. circle squaring. 2. cube duplication. 3. angle trisection.   Only in modern times, more than 2000 years after they were formulated, were all three ancient problems proved insoluble using only compass and straightedge.

Duplicating the cube: Duplicating the cube: constructing a cube whose volume is twice that of a given cube

Cube Duplication The problem appears in a Greek legend which tells how the Athenians, suffering under a plague, sought guidance from the Oracle at Delos as to how the gods could be appeased and the plague ended.

  The Oracle advised doubling the size of the altar to the god Apollo. The Athenians therefore built a new alter twice as big as the original in each direction and, like the original, cubical in shape However, as the Oracle (notorious for ambiguity and double-speaking in his prophecies) had advised doubling the size (i.e., volume), not linear dimension (i.e., scale), the new altar was actually eight times as big as the old one. As a result, the gods remained unappeased and the plague continued to spread unabated.

  The conics seem to have been discovered by Menaechmus (a Greek, c BC)   tutor to Alexander the Great.

Appollonius (c BC) (known as The Great Geometer)   Appollonius was the first to base the theory of all three conics on sections of one circular cone, right or oblique. He is also the one to give the name ellipse, parabola, and hyperbola.

Circle

Circle

Moves a circle of radius = 3 +7 in the x direction +4 in the y direction 7 4

Sketch the following circles

Find the centre and radius of each circle

Find the equation of the tangent to each circle at the point given.

Verify that are the parametric equations of the circle

A curve is drawn in the x, y plane so that any point on the curve is twice as far from (1, 0) as it is from (0, 0). Show that the coordinates of points on the curve satisfy

A point P moves so that its distance from the point (1, 4) is always three times its distance from the point (1, 0). Prove the locus of the point P is a circle. What is the radius and centre of this circle?

Homework: Exercises 13.1 and 13.2

For experts: Delta Exercise 4.5 Questions 11 onwards…