Important Terms and Properties of Hyperbola in Co-ordinate Geometry
The lecture covers essential terms related to hyperbola, including center, axes, vertices, latus-rectum, and focal property. It explains the focal property and its proof, defining hyperbola as the locus of a point with constant distances from fixed points.
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2 D Co-ordinate Geometry Lecture-19 The hyperbola Dated:-18.05.2020 PPT-15 UG (B.Sc., Part-1) Dr. Md. Ataur Rahman Guest Faculty Department of Mathematics M.L. Arya, College, Kasba PURNEA UNIVERSITY, PURNIA
Important terms related to an hyperbola Centre:- The mid-point of the line segment joining the foci of the ellipse is called centre. Here C- centre Axes of the hyperbola: Transverse axis:-The line segment passing through the foci of the hyperbola is called transverse axis. AA transverse axis and AA B K L M' M X A' Z' C A Z S' S(focus) (focus) K' L' B Directrix = 2 a length of transverse axis Conjugate axis:-The line segment passing through the centre and perpendicular to the conjugate axis of the hyperbola. In the figure BB Conjugate axis
Continue Vertices:-The points at which the hyperbola intersects the transverse axis of are called its vertices. In the above figure, A and A are vertices of the hyperbola. Latus-rectum:-The chord passing through the focus and perpendicular to the transverse axis. In the Figure, LL and KK are latus rectum . 2 2 b a = = = and Length of the latus rectum LL KK Note:- In an ellipse Length of transverse-axis Length of conjugate-axis = = 2 AA BB a = = 2 b
The focal property of a hyperbola Property:-The difference of the focal distances of any point on a hyperbola is constant and is equal to the length of the transverse axis of the hyperbola. i.e. If P be a point on the ellipse, then B M' M P(x,y) X A' Z' C A Z S(ae,0) S'(ae,0) B Directrix = = 2 ( a length of transverse axis ) PS PS AA
The focal property of a hyperbola Proof:- Let P(x, y) be any point on the hyperbola 2 2 1....(1) a b x y = 2 2 By the definition, PS PM = e a e = = = ex a ...(2) PS ePM e x a e = = + = ex a + ...(3) PS ePM e x (2) ex a ex a + (3), + = Subtracting PS PS from we get a = = = 2 AA Length of transverse axis
The focal property According to focal property Definition of a hyperbola:-A hyperbola is the locus of a point in a plane such that the difference of its distances from two fixed points in the plane is always constant and is equal to length of transverse-axis. i.e. tan PS PS cons t + = = 2 a
Summary Properties Transverse hyperbola 2 2 1, 0 a b b a e = Conjugate hyperbola x y b a b a e = x y 2 2 = b a = 1, 0 b a 2 2 2 2 ( ) ( ) 2 2 2 1 2 2 2 1 Centre Vertices Foci Length of transverse-axis Length of conjugate-axis (0,0) (0,0) (-a,0) and (a,0) (-ae,0) and (ae,0) (0,-a) and (0,a) (0,-ae) and (0,ae) 2a 2b 2a 2b Eq. of the transverse-axis Eq. of the conjugate-axis Length of the latus-rectum Y=0 X=0 X=0 2 2b a Y=0 2 2b a Eccentricity + 2 2 + 2 2 a b a b = = e e a a
