. . Ellipse We invoke that an ellipse is the locus of a point which moves such that its distance from a fixed point (focus) bears a constant ratio (eccentricity) less than unity its distance from its directrix bearing a constant ratio e ( < < e (i) Equation of an Ellipse in standard form Let S be a focus, l be a directrix, e be the eccentricity ( < < e ) and P x y ( , ) be the moving point. Draw SZ and PM perpendicular to l .
Let A and ′ A be the points which divide SZ internally and externally in the ratio e : respectively. Let AA ′ = . Let the point of intersection of the perpendicular bisector with AA ′ be C . Therefore CA and CA ′ = .
Choose C as origin and CZ produced as x -axis and the perpendicular bisector of AA ′ produced as y -axis. By definition, SA AZ = e SA A Z ' ' = e SA = eAZ SA ' = eA Z ' CA CS = e CZ CA A C CS ' = e A C CZ ' CS = e CZ ... ( ) CS = e a CZ ... ( ) ( ) + ( ) gives CZ e and ( ) − ( ) gives CS ae Therefore M is a e y and S is ae , ) .
By the definition of a conic, SP PM = e ⇒ SP e PM ⇒ x ae ) + = e e + which on simplification yields x e = . Since - e is a positive quantity, write b = a e − Taking ae = c b Hence we obtain the locus of P as x which is the equation of an ellipse in standard form and note that it is symmetrical about x and y axis. Definition . ( ) The line segment AA ′ is called the major axis of the ellipse and is of length a