Understanding Hyperbolas in Coordinate Geometry
Explore the concepts of hyperbolas in coordinate geometry, including definitions, equations, proofs, and focal properties. Delve into key points to grasp the essence of hyperbolic curves.
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2 D Co-ordinate Geometry Lecture-18 The hyperbola Dated:-16.05.2020 PPT-14 UG (B.Sc., Part-1) Dr. Md. Ataur Rahman Guest Faculty Department of Mathematics M.L. Arya, College, Kasba PURNEA UNIVERSITY, PURNIA
2 D Co-ordinate Geometry Lecture-18 The ellipse Dated:-16.05.2020 PPT-13 UG (B.Sc., Part-1) Dr. Md. Ataur Rahman Guest Faculty Department of Mathematics M.L. Arya, College, Kasba PURNEA UNIVERSITY, PURNIA
The hyperbola Definition:-A hyperbola is the Locus of a point which Moves in a plane such that the ratio of its distance from the fixed point (called focus) and from the fixed line (called directrix) is always constant and greater than unity. i.e. Where S and F are called foci ,ZM and Z'M' are called directrix and C is called centre of the hyperbola. B M' M P X Z' C Z F S(focus) (focus ) B Directrix PS PM = = tan 1 ( ) Cons t e eccentricity
Equation of the hyperbola a e Let S (ae,0) be the focus and be the eq. of the given directrix ZM of the ellipse, where e is the eccentricity. Let P(x, y) be any point on the hyperbola, then by the definition of the hyperbola, = x B M' M P 1 e X A' Z' A Z S' S(focus) (focus) B Directrix PS PM where AA = = .....(1) e PS ePM = = = 2 a CA CA a , , P draw PN ON = OZ and PM y = AZ then x and PN a e a e = = = = , ...(2) Now PM NZ OZ OM x as OZ
Proof Continue From (1),we get ( ) ( ) 2 2 = = PS ePM PS PM 2 a e ( ) ( ) 2 2 + = 2 0 x ae y e x 2 + a ex + + = 2 2 2 2 2 2 x aex a e y e e aex + + + = 2 2 2 2 2 2 2 2 2 x aex a e y a e x ( ) y ( ) = 2 2 2 2 2 1 1 x e y a e 2 2 x a + = 2 2 1 1 (1 ) 0 ( ) But e a e negativ e ( ) 2 2 2 1 a e 2 2 x a y e = 1 ( ) 2 2 2 1 a 2 2 x a y b 2 = ( e 1 tan ) 1 . S da rd equa tion of the transverse hyperbo l a 2 2 = 2 2 where b a
