Coordinate Geometry Rene Descartes, considered the father of modern philosophy (Cogito ergo sum), also had a great influence on mathematics. He and Fermat corresponded regularly and as a result of their interest in tangents to curves Descartes conceived the idea of coordinate geometry which bridges the gap between algebra and geometry. 1596-1650 A plane in which each point is identified with a ordered pair of real numbers (x,y) is called a coordinate (or Cartesian) plane.
We construct such a plane by drawing two perpendicular lines, one horizontal one vertical, called the coordinate axes that meet at a point called the origin.
Given any two points P(x1,y1) and Q(x2,y2), the distance between them is d(p,q) = (x2 - x1) 2 + (y2 - y1) 2 (Board) and the midpoint of the segment joining them has coordinates ( (x1 + x2)/2, (y1 + y2)/2 ) (Board) Example. Sketch the triangle with vertices A(1,1), B(2,4) and C(4,0). Show that it is an isosceles triangle and find its area. (The units are in inches.)
Definition: Given an equation in two variables, its graph is the set of all points in the plane whose coordinates satisfy the equation. Examples. 1. The point (3,4) is on the graph of x 2 + y 2 = 25 because if you replace x by 3 and y by 4, the result is a true statement: 3 2 + 4 2 = 25. 2. Similarly, the point (-7,0) is on the graph of 2x - y = -14. Definition: Any equation of the form y = mx + b where m and b are constants is called a linear equation; its graph is a straight line. Examples: y = 3x + 5, 3y = 6x - 9, y = 2 and x = -3 are all examples of linear equations. Their graphs are straight lines. Example: Find an equation of the line through (-1,1) and (1,2).
Definition: Given a point in the plane C(h,k), a circle with center C and radius r is the set of all points in the plane that are at a distance r from C. It follows from the distance formula, that an equation of the circle is (x - h) 2 + (y - k) 2 = r 2 Example: Find an equation of the circle with center (2,3) and radius 4. Definition: A line is tangent to a given circle if it touches the circle in exactly one point, called the point of tangency. Theorem: A tangent to a circle is perpendicular to the radius drawn from the center to the point of tangency. Proof?
Definition: Given a line d and a point F not on d, a parabola is the set of points in the plane that are equidistant from d and F (picture on the board.) The line d is the directrix and the point F is the focus. Parabolas occur in nature as the paths followed by comets that do not return. It is also the path of a projectile fired from a gun. If a light source is placed at the focus of a parabolic mirror, then the rays are reflected parallel to the axis to form a powerful beam. Example: Sketch the graph of 8x =y 2.
Definition: Given two distinct points F1 and F2, an ellipse is the set of all points in the plane whose distance from F1 and F2 have a constant sum. The two points are called foci and the midpoint between them is the center. (Picture on the board.) The orbits of the planets in our solar system are ellipses with the sun at one focus. Example: Sketch the graph of (x 2 /36) + (y 2 /9) = 1
All of the curves we have just discussed are called Conic Sections. Why is that? Conic sections have a long and rich history.
It is believed that the first definition of a conic section is due to Menaechmus (died 320 BC). His work did not survive and is only known through secondary accounts. Euclid ( c. 300 BC ) is said to have written four books on conics but these were lost. Archimedes (died c. 212 BC) is known to have studied conics, having determined the area bounded by a parabola and an ellipse. The only part of this work to survive is a book on the solids of revolution of conics. The greatest progress in the study of conics by the ancient Greeks is due to Apollonius of Perga (died c. 190 BC), whose eight-volume Conic Sections or Conics summarized and greatly extended existing knowledge. Apollonius's major innovation was to characterize a conic using properties within the plane and intrinsic to the curve; this greatly simplified analysis. With this tool, it was now possible to show that any plane cutting the cone, regardless of its angle, will produce a conic.
Apollonius's work was translated into Arabic (the technical language of the time) and much of his work only survives through the Arabic version. Persians found applications of the theory; the most notable of these was the Persian mathematician and poet Omar Khayyám who used conic sections to solve algebraic equations. Johannes Kepler (1571-1630) used conics to write out his laws of planetary motion. Girard Desargues and Blaise Pascal developed a theory of conics using an early form of projective geometry and this helped to provide impetus for the study of this new field. Meanwhile, René Descartes applied his newly created coordinate geometry to the study of conics. This had the effect of reducing the geometrical problems of conics to problems in algebra.
Video of Conic Sections
Exercises 1. Find an equation of the line through (1,3) and (-2,4) 2. Find an equation of the circle with (a) center (0,0) and radius 3 and (b) center (1,-3) and radius 5 3. Given a point A that is outside of given circle with center O. Draw a picture showing two lines through A that are each tangent to the circle at different points B and C. Prove that the line segments AB and AC are congruent. Hint. Prove that triangles AOB and AOC are congruent. 4. Sketch the graph of y = x 2 ; what is the name of this curve? 5. Sketch the graph of (x 2 /4) + (y 2 /9) = 1. What is this curve? NO CLASS NEXT WEEK