9172020 Ellipse 1 Chapter 13 Ellipse 9172020 Ellipse

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9/17/2020 Ellipse 1

9/17/2020 Ellipse 1

Chapter 13 Ellipse 椭圆 9/17/2020 Ellipse 2

Chapter 13 Ellipse 椭圆 9/17/2020 Ellipse 2

Definition: The locus of a point P which moves such that the ratio of

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

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).

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

(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

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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9/17/2020 Ellipse 8

Properties of ellipse 1. The curve is symmetrical about both axes. 2. 9/17/2020 Ellipse

Properties of ellipse 1. The curve is symmetrical about both axes. 2. 9/17/2020 Ellipse 9

3. 9/17/2020 Ellipse 10

3. 9/17/2020 Ellipse 10

y Q Q’ B’ Z A S O S’ A’ x Z’ B 9/17/2020

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

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

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

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

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

(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

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

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

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

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

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

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)

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,

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

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’,

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

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,

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

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

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

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

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

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

Soln: a=2 ; b=4/3 9/17/2020 Ellipse 34

9/17/2020 Ellipse 35

9/17/2020 Ellipse 35

y Q P O x N The angle QON is called the eccentric angle

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

Geometrical interpretation of the parameter 9/17/2020 Ellipse 37

Area of the ellipse 9/17/2020 Ellipse 38

Area of the ellipse 9/17/2020 Ellipse 38

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

The area of the ellipse is 4 times the area in the positive quadrant. 9/17/2020 Ellipse 39

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9/17/2020 Ellipse 40

e. g. 7 The semi-minor axis of an ellipse is of length k. If

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

Soln: 9/17/2020 Ellipse 42

Tangent and normal at the point to the ellipse 9/17/2020 Ellipse 43

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

We have Equation of tangent is : 9/17/2020 Ellipse 44

i. e. 9/17/2020 Ellipse 45

i. e. 9/17/2020 Ellipse 45

Equation of normal at is : i. e. 9/17/2020 Ellipse 46

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

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

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

At Q, 9/17/2020 Ellipse 49

The coordinates of the midpoint of PQ are : The required locus is 9/17/2020

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

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.

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

Hence, 9/17/2020 Ellipse 53

e. g. 9 For what values of c is the line y=1/2 x+c a

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 1 9/17/2020 Ellipse 55

Method 2 9/17/2020 Ellipse 56

Method 2 9/17/2020 Ellipse 56

Method 3 9/17/2020 Ellipse 57

Method 3 9/17/2020 Ellipse 57

9/17/2020 Ellipse 58

9/17/2020 Ellipse 58

e. g. 10 Find the equations of the tangent and normal to the ellipse

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

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

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

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

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

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

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’

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

Soln: Let L be Let M be L’ will be the point 9/17/2020 Ellipse 67

9/17/2020 Ellipse 68

9/17/2020 Ellipse 68

Hence, (LM)(L’M) is a constant. 9/17/2020 Ellipse 69

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

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

Eccentric angle of an ellipse Eccentric angle ? 13 b Q 17, 13 f Q 7 9/17/2020 Ellipse 71

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9/17/2020 Ellipse 72

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9/17/2020 Ellipse 74

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9/17/2020 Ellipse 75

13 f Q 9 M P(acosΦ, bsin Φ) O Eqn of tangent line PM

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

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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9/17/2020 Ellipse 78

13 f Q 5 y O 9/17/2020 x Ellipse 79

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

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

Hence, the angles are : 9/17/2020 Ellipse 81

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9/17/2020 Ellipse 82

2006 UEC Advanced Math Paper 2 9/17/2020 Ellipse 83

2006 UEC Advanced Math Paper 2 9/17/2020 Ellipse 83

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9/17/2020 Ellipse 84

A(1, 3) B X C 9/17/2020 Ellipse 85

A(1, 3) B X C 9/17/2020 Ellipse 85

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9/17/2020 Ellipse 87