Mathematics · Ch 5 — Two Dimensional Analytical Geometry-II
Introduction
Introduction
Analytical (coordinate) geometry describes geometric objects — points, lines, circles, parabolas, ellipses, hyperbolas — using algebra on a Cartesian coordinate system. Roughly two thousand years ago (around the 2nd–1st century BCE), the ancient Greeks already studied conic curves purely because they found the ideas exciting and challenging — they could not have imagined the uses later centuries would find for them.
Analytical methods for such geometric problems were systematically developed in the first half of the 17th century, chiefly by René Descartes, along with Fermat, Kepler, Newton, Euler, Leibniz, l'Hôpital, Clairaut, Cramer and the Jacobis. Analytic geometry grew out of the need to bring algebraic techniques to geometric problems, and its development has since reached deep into industry, medicine, and scientific research.
Johannes Kepler's laws of planetary motion — that every planet, including Earth, orbits the Sun on an ellipse with the Sun at one focus, obeying an inverse-square law — grew directly out of this coordinate approach and helped extend Euclidean geometry's reach into physics. Euler's systematic coordinate study of space curves and surfaces was later carried further by Einstein in the theory of relativity.
Where the Four Conics Show Up
- Circle — gears, dam vents, wheels, and circular geometry generally (a route into trigonometry too)
- Parabola — arches, satellite dishes, solar cookers, headlights, suspension bridges, and searchlights
- Ellipse — arches, medical lithotripsy, whispering galleries, Nd–YAG lasers, and gears
- Hyperbola — telescopes, cooling towers, and locating ships or aircraft
A Motivating Problem
A driver delivering a truck of books ( wide, high) sees two warning signs at a semi-elliptical tunnel entrance: "Tunnel is high at the centre peak" and "Tunnel is wide." Will the truck clear the archway? This chapter builds exactly the tools needed to answer that question — we return to it once the chapter is complete.
Learning Objectives
By the end of the chapter you should be able to:
- write the standard-form equations of a circle, parabola, ellipse and hyperbola;
- find the centre, vertices, foci, etc. from a conic's equation;
- derive the tangent and normal to each conic;
- classify a general second-degree equation's conic (including its degenerate forms);
- write conics in parametric form; and
- apply all of the above to real-life situations.
What this figure shows. Solar cooker dish, satellite receiver, arch bridge, whispering-gallery ceiling and a suspension-bridge cable — five everyday objects whose outlines are a circle, parabola or ellipse, used to motivate the chapter before any equation is written.
Figure 5.1–5.5: Conic applications collage.