Ellipses: Equations, Eccentricity & Co-ordinates
Solutions to ellipse problems including equations, eccentricity determination, and co-ordinate calculations. Solve for focus, directrix, latus rectum, and more in this detailed geometry lecture series.
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2 D Co-ordinate Geometry Lecture-16 The ellipse Dated:-13.05.2020 PPT-10 UG (B.Sc., Part-1) Dr. Md. Ataur Rahman Guest Faculty Department of Mathematics M.L. Arya, College, Kasba PURNEA UNIVERSITY, PURNIA
Problems (1). Find the equation of the ellipse whose focus (6,7) and directrix x+y+2=0 and its eccentricity is (2). Find the eccentricity of an ellipse if its latus-rectum is equal to one half of its major axis. (3) Find the co-ordinates of the centre, the foci, length of latus-rectum and the eccentricity of the ellipse 2 3 4 12 13 0. x y x y + + = 1 3 2 2 (4) Show that represents an ellipse. Also find its eccentricity. + + 1 0 + = 2 2 4 8 2 x y x y
Solution of (1) Solution:- Let S (6,7) be one focus, P(x, y) be any point on the ellipse and PM be its distance from the directrix By the definition of ellipse 1 3 3 3 PS PM PS PM = = 1 + + = = = 2 0...(1) x y and eccentricity e 3 PS PM = = e 2 2 2 + + 2 x y + + = 2 2 3 ( 6) ( 7) x y 2 2 1 1 + + + + + 2 2 4 2 2 + 4 4 x y xy y x + + + = 2 2 3 ( 12 36) ( 14 49) x x y y + + + = + + + + 2 2 2 2 6( 12 14 85) 4 2 4 4 x y y x y x x y = xy y x + + 2 2 5 5 2 76 88 506 0 x xy y
Solution of (2) Solution:-Let the eq. of the ellipse be 2 2 x a where length of major axis and length of major axis y b + = 1.....(1) where a b 2 2 = 2 a = 2 b 1 2 According to question = Latus rectum major axis 2 2 1 2 a e b a = = 2 2 2 2 .....(2) a b a = 2 2 2 (1 ) ) But b e = = 2 2 2 2 1 ( 2 1 a a e 1 = e ecce ntricity 2
Solution of (3) The Eq. of the ellipse is given by + + = = 2 2 2 3 x 4 12 13 4 ) 13 4 1 0....(1) + + x y x + + y = 2 2 2( x x x 2 ) 3( 1) 3( 3( ( 1 3 0 x y y y y + 4) 13 14 + = 2 2 2( 2( ( 2 0 x + 2 2 1) 2) y 2 2 1) 2) y + = 1....(2) 1 2 2 2 x y b + = (2) 1. Comparing witha 2 2 = ( 1, 2) (0,0) Centre x Centre y (1,2)
Solution continue Solution: 1 1 = = a and b 2 3 1 = = = 2 2 2 Length of major axis a 2 1 3 2 2 3 1 3 2 = = = min 2 2 Length of or axis b 3 2 2 b a = = Length of Latus rectum 1 2 2 2 1 a b = = = = Eccentricity e 1 a 3 2
