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DEFINITION A hyperbola is the locus of a point which moves in a plane such that the ratio of its distance from a fixed point and a given straight line is always constant and is greater than 1. The fixed point is called the focus, the fixed line is called the directrix and the constant ratio is called the eccentricity of the hyperbola and denoted by e. In the given figure, Sis the focus and NW the directrix. Let P be any point on the hyperbola, then N M Pp 90° See Zz s PM N Equation of a hyperbola can be obtained if the coordinates of its focus, equation of its directrix and eccentricity are given. GENERAL EQUATION OF HYPERBOLA Let (a, b) be the focus S, and x + my +n = 0 is the equation of firectrix. Let P(x, y) be any point on the hyperbola. Then by defination. => SP=ePM(e>1, isthe eccentricity) (@-a? +(y-6) P(x,y) = — ag (letmy +n) x (? +m’) 5 => (P +m?) (x-aP + (y-bY} S(ab) 3 =e {I+ my +n}? ° reduces to ax? + 2hxy + by? + 2gx + 2fy + c = 0, in which A#0, h?> ab. STANDARD EQUATION & DEFINITION(S) Let S be the focus & ZN is the directrix of an ellipse. Draw perpendicular from Sto the directrix which meet it at Z.A moving point is on the hyperbola such that PS=ePM then there is point lies on the line SZ and which divide $Z internaly yi at A and externally at 4’ in the ratioofe: 1. SA=eAZ Therefore Ai SA! = eA'Z i) Let AA’ = 2a & take C as mid point of AA’ CA=CA'=a Add (i) & (ii) SA+SA'=e(AZ+A'Z) (CS—CA)+(CA' + CS)= e [CA —CZ+ CA'+ CZ] 2CS=2e-CA CS=ae Subtract (ii) & (i), we get SA'-SA=e(A'Z-AZ) (CA'+ CS) —(CS— CA) = e [(CA' + CZ)—(CA-C)] 2CA=2e-CZ=>CzZ=". © Consider CZ line as x-axis, C as origin & perpendicular to this line & passes through C is considered as y-axis. Now ) represent important parameters on coordinates plane. Let P(x,’ is a moving point, then By definition of hyperbola. PS=ePM => (PS)=e2(PM? 2 = (-ae?+-o7=e(x-4] a 
