TEN - TOPICS IN ANALYTIC GEOMETRY

Abstract

There exist three special cases of the second degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.: the ellipse, the parabola, and the hyperbola. Each of these three curves plus the circle, which is a special kind of ellipse, can be obtained as the intersection of a plane with a right circular cone. That is why these curves are called conic sections. This chapter discusses these curves. The graph of most second degree equations is a circle, an ellipse, a parabola, or a hyperbola. The exceptional cases, called degenerate conic sections, consist of equations of which graphs are straight lines, single points, or empty. An ellipse is the set of points (x, y) such that the sum of the distances from (x, y) to two given points is fixed. Each of the two points is called a focus of the ellipse. The chapter presents the calculation of the equation of an ellipse with examples. A parabola is the set of points (x, y) equidistant from a fixed point and a fixed line that does not contain the fixed point. The fixed point is called the focus and the fixed line is called the directrix. The chapter discusses how to calculate the equation of a parabola with examples. A hyperbola is a set of points (x, y) with the property that the positive difference between the distances from (x, y) and each of two given (distinct) points is fixed. Each of the two given points is called a focus of the hyperbola. The chapter presents how to calculate the equation of a hyperbola with examples. The curves have had their axes parallel to the two coordinate axes. To obtain curves like an ellipse, a parabola, and a hyperbola, it is only necessary to rotate the coordinate axes through an appropriate angle.