9172020 Ellipse 1 Chapter 13 Ellipse 9172020 Ellipse
- Slides: 87
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Chapter 13 Ellipse 椭圆 9/17/2020 Ellipse 2
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. 9/17/2020 Ellipse 3
Q Y P(x, y) M A Z S a 9/17/2020 O A’ X a Ellipse 4
Let A, A’ divide SZ internally and externally in the ratio e: 1 (e<1). Then A, A’ are points on the ellipse. Let 9/17/2020 AA’=2 a SA=e. AZ; SA’=e. A’Z SA - SA’=e. AZ - e. A’Z =e(A’Z - AZ)=e. AA’ Ellipse 5
(a + OS)-(a – OS)=2 ae OS=ae i. e. Also, S is the point (-ae, 0) SA’ + SA = e(A’Z + AZ) 2 a=e(a +OZ + OZ – a) =2 e. OZ 9/17/2020 Ellipse 6
Let P(x, y) be any point on the ellipse. Then where 9/17/2020 PS=e. PM PM is perpendicular to ZQ. Ellipse 7
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Properties of ellipse 1. The curve is symmetrical about both axes. 2. 9/17/2020 Ellipse 9
3. 9/17/2020 Ellipse 10
y Q Q’ B’ Z A S O S’ A’ x Z’ B 9/17/2020 Ellipse 11
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. 9/17/2020 Ellipse 12
AA’ is the major axis, BB’ is the minor axis and O the centre of the ellipse. AA’=2 a ; BB’=2 b The eccentricity e of the ellipse is given by : 9/17/2020 Ellipse 13
e. g. 1 Find (i) the eccentricity, (ii) the coordinates of the foci, and (iii) the equations of the directrices of the ellipse 9/17/2020 Ellipse 14
Soln: (i) Comparing the equation with We have, 9/17/2020 a=3, b=2 Ellipse 15
(ii) Coordinates of the foci are (-ae, 0), (ae, 0) (iii) Equations of directrices are 9/17/2020 Ellipse 16
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. 9/17/2020 Ellipse 17
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. 9/17/2020 Ellipse 18
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 (x 1, y 1), those of the other end are (-x 1, -y 1). 9/17/2020 Ellipse 19
Equation of the tangent at the point (x’, y’) to the ellipse 9/17/2020 Ellipse 20
Differentiating w. r. t x, Gradient of tangent at (x’, y’) is 9/17/2020 Ellipse 21
Equation of tangent at (x’, y’) is : 9/17/2020 Ellipse 22
e. g. 3 Find the equation of the tangent at the point (2, 3) to the ellipse. Soln: 9/17/2020 Ellipse 23
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 9/17/2020 Ellipse 24
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. 9/17/2020 Ellipse 25
Soln: Let P be (x’, y’), A be (x’, 0) Hence, M is (x’, y’/2) P is on the circle, M P 1 A Because M coordinates are x=x’ and y=y’/2. Put x’=x and y’=2 y into eqn 1 Locus is 9/17/2020 Ellipse 26
Locus formed by the above example. 9/17/2020 Ellipse 27
Standard equation vertices (-a, 0), (0, b), (0, -b) (-b, 0), (0, -a), (0, a) foci (-ae, 0), (ae, 0) (0, -ae), (0, ae) 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) directrices 9/17/2020 Ellipse 28
y y a b a’ F O F F’ a x b’ O b x F’ b’ a’ 9/17/2020 Ellipse 29
Parametric equations of an ellipse 9/17/2020 Ellipse 30
is always satisfied by the values : 长轴 Major axis is a parameter 9/17/2020 Ellipse Minor axis 短轴 31
The parametric coordinates of any point on the curve are : 9/17/2020 Ellipse 32
e. g. 6 Find the parametric coordinates of any point on each of the following ellipses: 9/17/2020 Ellipse 33
Soln: a=2 ; b=4/3 9/17/2020 Ellipse 34
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y Q P O x N The angle QON is called the eccentric angle of P. The circle is called the auxiliary circle of the ellipse. 9/17/2020 Ellipse 36
Geometrical interpretation of the parameter 9/17/2020 Ellipse 37
Area of the ellipse 9/17/2020 Ellipse 38
The area of the ellipse is 4 times the area in the positive quadrant. 9/17/2020 Ellipse 39
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e. g. 7 The semi-minor axis of an ellipse is of length k. If the area of the ellipse is , find its eccentricity. 9/17/2020 Ellipse 41
Soln: 9/17/2020 Ellipse 42
Tangent and normal at the point to the ellipse 9/17/2020 Ellipse 43
We have Equation of tangent is : 9/17/2020 Ellipse 44
i. e. 9/17/2020 Ellipse 45
Equation of normal at is : i. e. 9/17/2020 Ellipse 46
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. 9/17/2020 Ellipse 47
y Soln: P O x Q P’ Let Eqn of diameter OP’ is Eqn of normal at P is 9/17/2020 Ellipse 48
At Q, 9/17/2020 Ellipse 49
The coordinates of the midpoint of PQ are : The required locus is 9/17/2020 Ellipse 50
Equation of a tangent in terms of its gradient to the ellipse 9/17/2020 Ellipse 51
The equation of the tangent at the point to the ellipse is : i. e. Writing the gradient 9/17/2020 Ellipse 52
Hence, 9/17/2020 Ellipse 53
e. g. 9 For what values of c is the line y=1/2 x+c a tangent to the ellipse ? 3 methods to do this Q! 9/17/2020 Ellipse 54
Method 1 9/17/2020 Ellipse 55
Method 2 9/17/2020 Ellipse 56
Method 3 9/17/2020 Ellipse 57
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e. g. 10 Find the equations of the tangent and normal to the ellipse at the point. 9/17/2020 Ellipse 59
e. g. 11 Find the equations of the tangents to the ellipse which are parallel to the diameter y=2 x. 9/17/2020 Ellipse 60
Soln: The tangent has gradient m=2. Because 9/17/2020 Ellipse 61
e. g. 12 Find the locus of the point of intersection of perpendicular tangents to the ellipse. 9/17/2020 Ellipse 62
Soln: Let be the pt. of intersection of a pair of perpendicular tangents. 9/17/2020 Ellipse 63
Product of roots, As the tangents are perpendicular, The locus is a circle The circle is called the director circle of the ellipse. 9/17/2020 Ellipse 64
Ex 13 d and Ex 13 e Abandon them! Ex 13 f do only Q 1, Q 3, Q 5, Q 7, Q 9 9/17/2020 Ellipse 65
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. 9/17/2020 Ellipse 66
Soln: Let L be Let M be L’ will be the point 9/17/2020 Ellipse 67
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Hence, (LM)(L’M) is a constant. 9/17/2020 Ellipse 69
Conclusion: In analytic geometry, the ellipse is represented by the implicit equation : Or of the form : 9/17/2020 Ellipse 70
Eccentric angle of an ellipse Eccentric angle ? 13 b Q 17, 13 f Q 7 9/17/2020 Ellipse 71
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13 f Q 9 M P(acosΦ, bsin Φ) O Eqn of tangent line PM is : 1 9/17/2020 Ellipse 76
So, gradient of PM is So, gradient of OM is Eqn of line OM is : Sub. into 1 : 9/17/2020 Ellipse 77
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13 f Q 5 y O 9/17/2020 x Ellipse 79
OS=ae=1 OQ= y Q 1 S’ 9/17/2020 O 3 2 P x S Ellipse 80
Hence, the angles are : 9/17/2020 Ellipse 81
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2006 UEC Advanced Math Paper 2 9/17/2020 Ellipse 83
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A(1, 3) B X C 9/17/2020 Ellipse 85
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