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® The eccentricity of a hyperbola is the ratio of the distances from the centre of the hyperbola to one of the foci and to one of the vertices of the hyperbola. Historical Note Geometry is one of the most ancient branches of mathematics. The Greek geometers investigated the properties of many curves that have theoretical and practical importance. Euclid wrote his treatise on geometry around B.C.

He was the first who organised the geometric figures based on certain axioms suggested by physical considerations. Geometry as initially studied by the ancient Indians and Greeks, who made essentially no use of the process of algebra. The synthetic approach to the subject of geometry as given by Euclid and in Sulbasutras , etc., was continued for some years. In the B.C., Apollonius wrote a book called ‘ The Conic ’ which was all about conic sections with many important discoveries that have remained unsurpassed for eighteen centuries.

Modern analytic geometry is called ‘ Cartesian ’ after the name of Rene Descartes ( - ) whose relevant ‘La Geometrie’ was published in . But the fundamental principle and method of analytical geometry were already discovered by Pierre de Fermat ( - ). Unfortunately, Fermats treatise on the subject, entitled Ad Locus Planos et So LIDOS Isagoge (Introduction to Plane and Solid Loci) was published only posthumously in . So, Descartes came to be regarded as the unique inventor of the analytical geometry.

Isaac Barrow avoided using cartesian method. Newton used method of undetermined coefficients to find equations of curves. He used several types of coordinates including polar and bipolar. Leibnitz used the terms ‘ abscissa ’, ‘ordinate’ and ‘coordinate’.

L’ Hospital (about ) wrote an important textbook on analytical geometry. Clairaut ( ) was the first to give the distance formula although in clumsy form. He also gave the intercept form of the linear equation. Cramer ( ) CONIC SECTIONS made formal use of the two axes and gave the equation of a circle as ( y – a ) + ( b – x ) = r He gave the best exposition of the analytical geometry of his time.

Monge ( ) gave the modern ‘point-slope’ form of equation of a line as y – y ′ = a ( x – x ′ ) and the condition of perpendicularity of two lines as aa ′ + = . S.F. Lacroix ( – ) was a prolific textbook writer, but his contributions to analytical geometry are found scattered. He gave the ‘two-point’ form of equation of a line as β β β = α ) α α – y – x – – ′ ′ and the length of the perpendicular from ( α , β ) on y = ax + b as ( β – a – b His formula for finding angle between two lines was tan θ a – a aa ′ =  ′  .

It is, of course, surprising that one has to wait for more than years after the invention of analytical geometry before finding such essential basic formula. In , C. Lame, a civil engineer, gave m E + m ′ E ′ = as the curve passing through the points of intersection of two loci E = and E ′ = . Many important discoveries, both in Mathematics and Science, have been linked to the conic sections.

The Greeks particularly Archimedes ( – B.C.) and Apollonius ( B.C.) studied conic sections for their own beauty. These curves are important tools for present day exploration of outer space and also for research into behaviour of atomic particles.