Ellipse Geometry: Important Terms and Focal Property
Explore important concepts and properties related to ellipses, including terms like centre, major axis, minor axis, vertices, and the focal property. Understand the geometric characteristics and focal property of ellipses through detailed explanations and visual representations.
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2 D Co-ordinate Geometry Lecture-14 The ellipse Dated:-11.05.2020 PPT-06 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 ellipse Centre: The mid-point of the line segment joining the foci of the ellipse. Here O- centre Axes of the ellipse: Major axis:-The line segment passing the foci and whose end points on the ellipse. AA major axis and AA P(x,y) B L K M M Z O Z A centre S(focus) S (focus) A K' L' B Directrix Directrix length of major axis = 2 a Minor axis:-The line segment passing through the centre and perpendicular to the major- axis with end points on the ellipse. In the figure BB Minor-axis
Continue Vertices:-The end points of the major-axis of an ellipse are called its vertices. In the given figure, A and A are vertices of the ellipse. Latus-rectum:-The chord passing through the focus and perpendicular to the major-axis. In the Figure, Note:- In an ellipse Length of major-axis Length of semi-major axis Length of minor-axis Length of semi-minor axis = LL and KK are latus rectum . = = = 2 AA a = b = = OA = OA a = 2 BB = OB OB b
The focal property of an ellipse Property:-The sum of the focal distances of any point on an ellipse is constant and equal to the length of the major axis of the ellipse. B P(x,y) M M Z X O N S(focus) Z A S (focus) A i.e. If P be a point on the ellipse, then B Directrix + = = 2 ( a length of major axis ) PS PS AA
The focal property of an ellipse Proof:- Let P(x, y) be any point on the ellipse 2 2 1....(1) a b x y + = 2 2 By the definition, a e = = a ex = ...(2) PS ePM e x a e = = + a ex = + ...(3) PS ePM e x (2) = (3), + + Adding PS + and a ex a ex we get = = = 2 PS a AA Length of major axis
The focal property According to focal property Definition of an ellipse:-An ellipse is the locus of a point in a plane such that the sum of its distances from two fixed points in the plane is always constant and equal to length of major- axis. i.e. tan 2 PS PS cons t + = = a
Summary Properties Horizontal ellipse 2 2 1, 0 b a = Vertical ellipse x y b a b a = x a b y 2 2 + = b a + = 1, 0 b a 2 2 2 2 ( ) ( ) 2 2 2 1 e 2 2 2 1 e Centre Vertices Foci Length of major-axis Length of minor-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 major-axis Eq. of the minor-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
