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2. Chapter 13
Ellipse 椭圆
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3. Definition:
The locus of a point P which moves such that the ratio of its distances from a fixed point S and from a fixed straight line ZQ is constant and less than one.
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4. Q Y
P(x,y) M A O X Z S A’ a a
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5. Then A, A’ are points on the ellipse.
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’
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6. Also, SA’ + SA = e(A’Z + AZ)
(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
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7. Let P(x,y) be any point on the ellipse.
Then PS=ePM
where PM is perpendicular to ZQ.
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9. 1. The curve is symmetrical about both axes.
Properties of ellipse
1. The curve is symmetrical about both axes. 2.
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10. 3.
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11. y Q Q’ B’ x Z A O A’ S’ S Z’ B
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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 .
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13. The eccentricity e of the ellipse is given by :
AA’ is the major axis, BB’ is the minor axis and O the centre of the ellipse.
AA’=2a ; BB’=2b
The eccentricity e of the ellipse is given by :
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14. e.g. 1
Find (i) the eccentricity, (ii) the coordinates of the foci, and (iii) the equations of the directrices of the ellipse
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15. (i) Comparing the equation with
Soln:
(i) Comparing the equation with
We have, a=3, b=2
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16. (ii) Coordinates of the foci are (-ae,0), (ae,0)
(iii) Equations of directrices are
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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.
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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.
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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).
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20. Equation of the tangent at the point (x’,y’) to the ellipse
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21. Gradient of tangent at (x’,y’) is
Differentiating w.r.t x,
Gradient of tangent at (x’,y’) is
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22. Equation of tangent at (x’,y’) is :
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23. e.g. 3 Find the equation of the tangent at the point (2,3) to the ellipse .
Soln:
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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
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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.
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26. 1 P Soln: M Let P be (x’,y’), A be (x’,0) A Hence, M is (x’,y’/2)
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
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27. Locus formed by the above example.
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28. x-axis, y-xis foci (-ae,0), (ae,0) Standard equation vertices
(-a,0),(a,0),(0,b),(0,-b)
(-b,0),(b,0),(0,-a),(0,a)
foci
(-ae,0), (ae,0)
(0,-ae), (0,ae)
directrices
Symmetry axes
x-axis, y-xis
Length of axes
semi major axis=a (x)
semi minor axis=b (y)
semi major axis=a (y)
semi minor axis=b (x)
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29. y y a b F a’ x x O a O F F’ b b’ F’ b’ a’
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30. Parametric equations of an ellipse
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31. is always satisfied by the values :长轴
Major axis
Minor axis
is a parameter 短轴
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32. The parametric coordinates of any point on the curve are :
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33. e.g. 6
Find the parametric coordinates of any point on each of the following ellipses:
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34. Soln:
a=2 ; b=4/3
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36. x y O The angle QON is called the eccentric angle of P.
The circle is called the auxiliary circle of the ellipse.
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37. Geometrical interpretation of the parameter
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38. Area of the ellipse
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39. The area of the ellipse is 4 times the area in the positive quadrant.
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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.
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42. Soln:
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43. Tangent and normal at the point to the ellipse
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44. Equation of tangent is :
We have
Equation of tangent is :
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45. i.e.
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46. Equation of normal at is :
i.e.
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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.
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48. Soln: Let y Eqn of diameter OP’ is Eqn of normal at P is P x P’ O Q
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49. At Q,
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50. The coordinates of the midpoint of PQ are :
The required locus is
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51. Equation of a tangent in terms of its gradient to the ellipse
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52. i.e. The equation of the tangent at the point to the ellipse is :
Writing the gradient
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53. Hence,
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54. e.g. 9
For what values of c is the line y=1/2x+c a tangent to the ellipse ?
3 methods to do this Q!
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55. Method 1
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56. Method 2
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57. Method 3
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59. e.g. 10 Find the equations of the tangent and normal to the ellipse
at the point
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60. e.g. 11 Find the equations of the tangents to the ellipse which are
parallel to the diameter y=2x.
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61. The tangent has gradient m=2.
Soln:
The tangent has gradient m=2.
Because
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62. e.g. 12
Find the locus of the point of intersection of perpendicular tangents to the ellipse .
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63. Let be the pt. of intersection of a pair of perpendicular tangents.
Soln:
Let be the pt. of intersection of a pair of perpendicular tangents.
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64. As the tangents are perpendicular,
Product of roots,
As the tangents are perpendicular,
The locus is a circle
The circle is called the director circle of the ellipse.
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65. Abandon them! Ex13f do only Q1, Q3, Q5, Q7, Q9 Ex13d and Ex13e
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66. e.g. 13
The extremities of any diameter of an ellipse are L, L’ and M is any other point on the curve. Prove that the product of the gradients of the chords LM, L’M is constant.
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67. Soln:
Let L be
Let M be
L’ will be the point
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69. Hence, (LM)(L’M) is a constant.
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70. Conclusion: Or of the form :
In analytic geometry, the ellipse is represented by the implicit equation :
Or of the form :
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71. Eccentric angle of an ellipse
13b Q17, 13f Q7
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76. 1 13f Q9 Eqn of tangent line PM is : M P(acosΦ,bsin Φ) O 9/13/2018
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77. Eqn of line OM is : Sub. into 1 : So, gradient of PM is
So, gradient of OM is
Eqn of line OM is :
Sub. into 1 :
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79. y
13f Q5 x O
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80. OS=ae=1 y
OQ= Q P 1 x S’ O S 3 2
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81. Hence, the angles are :
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83. 2006 UEC Advanced Math Paper 2
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85. A(1,3) X B C
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