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Ellipse: Definition, Properties, and Equations

Learn about the definition and properties of an ellipse, including its equation, foci, directrices, tangents, and normals. Discover how to find the area of an ellipse and the parametric equations for points on the curve.

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Ellipse: Definition, Properties, and Equations

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  1. Ellipse

  2. Chapter 13 Ellipse 椭圆 Ellipse

  3. Definition: The locus of a point P which moves such that the ratio of its distances from a fixed point Sand from a fixed straight line ZQ is constant and less than one. Ellipse

  4. Q Y P(x,y) M A O X Z S A’ a a Ellipse

  5. Let A, A’ divide SZ internally and externally in the ratio e:1 (e<1). Then A, A’ are points on the ellipse. Let AA’=2a SA=eAZ; SA’=eA’Z SA - SA’=eAZ - eA’Z =e(A’Z - AZ)=eAA’ Ellipse

  6. (a + OS)-(a – OS)=2ae OS=ae i.e. S is the point (-ae,0) Also, SA’ + SA = e(A’Z + AZ) 2a=e(a +OZ + OZ – a) =2eOZ Ellipse

  7. Let P(x,y) be any point on the ellipse. Then PS=ePM where PM is perpendicular to ZQ. Ellipse

  8. Ellipse

  9. Properties of ellipse 1. The curve is symmetrical about both axes. 2. Ellipse

  10. 3. Ellipse

  11. y Q Q’ B’ x Z A O A’ S’ S Z’ B Ellipse

  12. The above diagram represents a standard ellipse. The foci S, S’ are the points (-ae,0) , (ae,0). The directrices ZQ, Z’Q’ are lines x=-a/e, x=a/e . Ellipse

  13. AA’ is the major axis, BB’ is the minor axis and O the centre of the ellipse. AA’=2a ; BB’=2b The eccentricitye of the ellipse is given by : Ellipse

  14. e.g. 1 Find (i) the eccentricity, (ii) the coordinates of the foci, and (iii) the equations of the directrices of the ellipse Ellipse

  15. Soln: (i) Comparing the equation with We have, a=3, b=2 Ellipse

  16. (ii) Coordinates of the foci are (-ae,0), (ae,0) (iii) Equations of directrices are Ellipse

  17. e.g. 2 The centre of an ellipse is the point (2,1). The major and minor axes are of lengths 5 and 3 units and are parallel to the y and x axes respectively. Find the equation of the ellipse. Ellipse

  18. Soln: Centre of ellipse is (2,1). So (x-2) and (y-1). The major axis is parallel to y axis, the equation is Where b=3/2, a=5/2 i.e. Ellipse

  19. Diameters A chord of an ellipse which passes thru’ the centre is called a diameter. By symmetry, if the coordinates of one end of a diameter are (x1,y1), those of the other end are (-x1,-y1). Ellipse

  20. Equation of the tangent at the point (x’,y’) to the ellipse Ellipse

  21. Differentiating w.r.t x, Gradient of tangent at (x’,y’) is Ellipse

  22. Equation of tangent at (x’,y’) is : Ellipse

  23. e.g. 3 Find the equation of the tangent at the point (2,3) to the ellipse . Soln: Ellipse

  24. e.g. 4 Write down the equation of the tangent at the point (-2,-1) to the ellipse . Soln: Eqn of tangent at (-2,-1) is Ellipse

  25. e.g.5 Find the equation of the locus of the mid-point of a perpendicular line drawn from a point on the circle, , to x-axis. Ellipse

  26. P Soln: M Let P be (x’,y’), A be (x’,0) A Hence, M is (x’,y’/2) 1 P is on the circle, Because M coordinates are x=x’ and y=y’/2 . Put x’=x and y’=2y into eqn 1 Locus is Ellipse

  27. Locus formed by the above example. Ellipse

  28. Ellipse

  29. y y a b F a’ x x O a O F F’ b b’ F’ b’ a’ Ellipse

  30. Parametric equations of an ellipse Ellipse

  31. is always satisfied by the values : 长轴 Major axis Minor axis is a parameter 短轴 Ellipse

  32. The parametric coordinates of any point on the curve are : Ellipse

  33. e.g. 6 Find the parametric coordinates of any point on each of the following ellipses: Ellipse

  34. Soln: a=2 ; b=4/3 Ellipse

  35. Ellipse

  36. y Q P x O N The angle QON is called the eccentric angle of P. The circle is called the auxiliary circle of the ellipse. Ellipse

  37. Geometrical interpretation of the parameter Ellipse

  38. Area of the ellipse Ellipse

  39. The area of the ellipse is 4 times the area in the positive quadrant. Ellipse

  40. Ellipse

  41. e.g. 7 The semi-minor axis of an ellipse is of length k. If the area of the ellipse is , find its eccentricity. Ellipse

  42. Soln: Ellipse

  43. Tangent and normal at the point to the ellipse Ellipse

  44. We have Equation of tangent is : Ellipse

  45. i.e. Ellipse

  46. Equation of normal at is : i.e. Ellipse

  47. e.g. 8 PP’ is a double ordinate of the ellipse . The normal at P meets the diameter through P’ at Q. Find the locus of the midpoint of PQ. Ellipse

  48. y Soln: P Let x O Q P’ Eqn of diameter OP’ is Eqn of normal at P is Ellipse

  49. At Q, Ellipse

  50. The coordinates of the midpoint of PQ are : The required locus is Ellipse

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