. . Hyperbola We invoke that a hyperbola is the locus of a point which moves such that its distance from a fixed point (focus) bears a constant ratio (eccentricity) greater than unity its distance from its directrix, bearing a constant ratio e ( e > . (i) Equation of a Hyperbola in standard form with centre at ( , ) Let S be a focus, l be the directrix line, 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 .
Then CA = CA ' = a Choose C as origin and the line CZ produced as x -axis and the perpendicular bisector of AA ′ as y -axis. By definition, AS AZ e ′ ′ A S A Z e . Fig. .
Fig. . C x' B' γ' B A' P ( x,y ) B Z B′ S ( ae, ) S′ ( –ae, ) M L A ( a, ) C ( , ) Z′ A′ (– a, ) L′ l Vertices Foci Latus rectum Centre ⇒ AS eAZ ′ ′ A S eA Z ⇒ CS CA e CA CZ ′ ′ A C CS e A C CZ ⇒ CS e a CZ ) … ( ) CS e a CZ ) … ( ) ( ) ( ) gives CS ae and ( ) ( ) gives CZ e Hence, the coordinates of S are ( , ) ae . Since PM e , the equation of directrix is x e = .
Let P x y ( , ) be any point on the hyperbola. By the definition of a conic, SP PM e ⇒ SP e PM Then ( ae = e e ⇒ ae = ( ex ⇒ e = a e ⇒ x e ) = . Since e e >