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Chapter 13 Ellipse 椭圆 9/13/2018 Ellipse

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. 9/13/2018 Ellipse

Q Y P(x,y) M A O X Z S A’ a a 9/13/2018 Ellipse

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’ 9/13/2018 Ellipse

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 9/13/2018 Ellipse

Let P(x,y) be any point on the ellipse. Then PS=ePM where PM is perpendicular to ZQ. 9/13/2018 Ellipse

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1. The curve is symmetrical about both axes. Properties of ellipse 1. The curve is symmetrical about both axes. 2. 9/13/2018 Ellipse

3. 9/13/2018 Ellipse

y Q Q’ B’ x Z A O A’ S’ S Z’ B 9/13/2018 Ellipse

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 . 9/13/2018 Ellipse

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 : 9/13/2018 Ellipse

e.g. 1 Find (i) the eccentricity, (ii) the coordinates of the foci, and (iii) the equations of the directrices of the ellipse 9/13/2018 Ellipse

(i) Comparing the equation with Soln: (i) Comparing the equation with We have, a=3, b=2 9/13/2018 Ellipse

(ii) Coordinates of the foci are (-ae,0), (ae,0) (iii) Equations of directrices are 9/13/2018 Ellipse

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. 9/13/2018 Ellipse

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. 9/13/2018 Ellipse

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). 9/13/2018 Ellipse

Equation of the tangent at the point (x’,y’) to the ellipse 9/13/2018 Ellipse

Gradient of tangent at (x’,y’) is Differentiating w.r.t x, Gradient of tangent at (x’,y’) is 9/13/2018 Ellipse

Equation of tangent at (x’,y’) is : 9/13/2018 Ellipse

e.g. 3 Find the equation of the tangent at the point (2,3) to the ellipse . Soln: 9/13/2018 Ellipse

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 9/13/2018 Ellipse

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. 9/13/2018 Ellipse

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 9/13/2018 Ellipse

Locus formed by the above example. 9/13/2018 Ellipse

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) 9/13/2018 Ellipse

y y a b F a’ x x O a O F F’ b b’ F’ b’ a’ 9/13/2018 Ellipse

Parametric equations of an ellipse 9/13/2018 Ellipse

is always satisfied by the values : 长轴 Major axis Minor axis is a parameter 短轴 9/13/2018 Ellipse

The parametric coordinates of any point on the curve are : 9/13/2018 Ellipse

e.g. 6 Find the parametric coordinates of any point on each of the following ellipses: 9/13/2018 Ellipse

Soln: a=2 ; b=4/3 9/13/2018 Ellipse

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x y O The angle QON is called the eccentric angle of P. The circle is called the auxiliary circle of the ellipse. 9/13/2018 Ellipse

Geometrical interpretation of the parameter 9/13/2018 Ellipse

Area of the ellipse 9/13/2018 Ellipse

The area of the ellipse is 4 times the area in the positive quadrant. 9/13/2018 Ellipse

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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. 9/13/2018 Ellipse

Soln: 9/13/2018 Ellipse

Tangent and normal at the point to the ellipse 9/13/2018 Ellipse

Equation of tangent is : We have Equation of tangent is : 9/13/2018 Ellipse

i.e. 9/13/2018 Ellipse

Equation of normal at is : i.e. 9/13/2018 Ellipse

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. 9/13/2018 Ellipse

Soln: Let y Eqn of diameter OP’ is Eqn of normal at P is P x P’ O Q 9/13/2018 Ellipse

At Q, 9/13/2018 Ellipse

The coordinates of the midpoint of PQ are : The required locus is 9/13/2018 Ellipse

Equation of a tangent in terms of its gradient to the ellipse 9/13/2018 Ellipse

i.e. The equation of the tangent at the point to the ellipse is : Writing the gradient 9/13/2018 Ellipse

Hence, 9/13/2018 Ellipse

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! 9/13/2018 Ellipse

Method 1 9/13/2018 Ellipse

Method 2 9/13/2018 Ellipse

Method 3 9/13/2018 Ellipse

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e.g. 10 Find the equations of the tangent and normal to the ellipse at the point . 9/13/2018 Ellipse

e.g. 11 Find the equations of the tangents to the ellipse which are parallel to the diameter y=2x. 9/13/2018 Ellipse

The tangent has gradient m=2. Soln: The tangent has gradient m=2. Because 9/13/2018 Ellipse

e.g. 12 Find the locus of the point of intersection of perpendicular tangents to the ellipse . 9/13/2018 Ellipse

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. 9/13/2018 Ellipse

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. 9/13/2018 Ellipse

Abandon them! Ex13f do only Q1, Q3, Q5, Q7, Q9 Ex13d and Ex13e 9/13/2018 Ellipse

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. 9/13/2018 Ellipse

Soln: Let L be Let M be L’ will be the point 9/13/2018 Ellipse

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Hence, (LM)(L’M) is a constant. 9/13/2018 Ellipse

Conclusion: Or of the form : In analytic geometry, the ellipse is represented by the implicit equation : Or of the form : 9/13/2018 Ellipse

Eccentric angle of an ellipse 13b Q17, 13f Q7 9/13/2018 Ellipse

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1 13f Q9 Eqn of tangent line PM is : M P(acosΦ,bsin Φ) O 9/13/2018 Ellipse

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 : 9/13/2018 Ellipse

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y 13f Q5 x O 9/13/2018 Ellipse

OS=ae=1 y OQ= Q P 1 x S’ O S 3 2 9/13/2018 Ellipse

Hence, the angles are : 9/13/2018 Ellipse

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2006 UEC Advanced Math Paper 2 9/13/2018 Ellipse

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A(1,3) X B C 9/13/2018 Ellipse

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