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Pure Maths/Ellipse/p.1 Ellipse For the standard form of the equation of the ellipse, the focus is (ae , 0) and the directrix is x = Let P(x, y) be a variable point on the ellipse. y Distance of (x, y) from the focus (ae, 0) a Distance of (x, y) from the directrix x = e or = e Distance of (x, y) from the focus (- ae, 0) a Distance of (x, y) from the directrix x = e (-a, 0) x x = e x - ae a + y2 = e2 - x e 2 x2 1 - e 2 + y2 = a 2 1 - e 2 x2 y2 + a2 a 2 1 - e2 = 1 a e A’ (-a,0) (a, 0) x x F(ae, 0) Principal focus x=a e x=a y e B(0,b) x2 y2 + = 1 2 2 a b where b2 = a 2 1 - e 2 x a e x=x=- 0 F’(-ae,0) Secondary focus where 0 < e < 1. 2 a . e x F’(-ae,0) P x A(a, 0) x F(ae, 0) x . B’(0,-b) a a Consider PF + PF’ = e - x e x - - = 2a e e ∴The locus of a variable point moving in such a way that the sum of its distances from 2 fixed points (foci) is a constant is an ellipse. Remarks 1. A, A’, B, B’ - vertices of the ellipse ; AA’ - major axis x2 y2 + = 1 32 22 origin 0 BB’ - minor axis ; y - centre of the ellipse y x2 y2 + = 1 22 32 x x Pure Maths/Ellipse/p.2 2. The chord through the focus and perpendicular to the major axis - latus rectum (rectara) y (ae)2 y2 + = 1 a2 b2 L b4 y2 = b 2 1 - e 2 = a2 ∴ 3. 2b 2 length of latus rectum = LL' 2b2 1 = = AF a a(1 - e) x F(ae, 0) a 2a 2 1 - e2 a 2 1 - e A x L’ = 2 (1 + e) < 4 Point of intersection of the major axis and minor axis is called the centre of the ellipse. Any chord through the centre is called a diameter . One half of a diameter is called a semi-diameter . 4. The circle described on the major axis of an ellipse as diameter is known as the auxiliary circle of the ellipse. x 2 + y 2 = a 2 5. y Parametric Equation : (a) Trigonometric Parametric Equations auxiliary circle : x2 + y2 = a 2 a x2 y2 + = 1 a2 b2 x = a cos y = b sin x P 0 where is the eccentric angle and 0 < 2 . (b) Algebraic Parametric Equations x = a cos y = b sin Let t = tan . 2 a 1 - t2 x = 1 + t2 ; y = 2bt 1 + t2 Example 1. x2 y2 A circle cuts the ellipse + = 1 at the points A, B, C and D. a2 b2 Show that AB, CD are equally inclined to the x-axis. Let the equation of AB be y - mx - c = 0 and the equation of CD be y - m’x - c’ = 0. x Pure Maths/Ellipse/p.3 ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ 2. x2 y2 + = 1 . Variable line through R is drawn 2 2 a b y meeting the ellipse at two points P and Q. x R(h. k) Find the locus of the mid-point of the chord PQ. Let R(h, k) be an external point of the ellipse P ________________________________________________ x Q ________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ ________________________________________________________________________ Pure Maths/Ellipse/p.4 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ _______________________________________________________________________ Classwork (91) Chords with slope equal to 1 are drawn in the ellipse 2x2 + 2xy + y2 = 1 . Prove that the mid-points of those chords lie on a straight line, and find the equation of this line. (Answer : 3x + 2y = 0 ) ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ Tangents of an Ellipse The equation of tangent to the ellipse x2 y2 + = 1 at the point (x1 , y1 ) or (a cos , b sin ) is a2 b2 xx1 yy + 1 = 1 a2 b2 or x cos y sin + = 1 . a b Given a line y = mx + c , what is the condition for the line to be a tangent to the ellipse Note : If the line is vertical, e.g. x = c for some constant c, x2 y2 + = 1 ? a2 b2 Pure Maths/Ellipse/p.5 when c = a , it is a tangent to the ellipse when c > a , no intersection when c < a , two distinct points of intersection x2 y2 + = 1 a2 b2 y = mx + c b2 x2 + a 2(mx + c)2 = b2a 2 b2 + a 2m2 x2 + 2a 2mcx + a 2c2 - b2a 2 = 0 The line y = mx + c is a tangent to the ellipse ∴ x2 y2 + = 1 a2 b2 iff △ = 0 4a 4m 2c2 - 4 b2 + a 2m 2 a 2c2 - b2a 2 = 0 a 2 m 2 c2 - b 2 c 2 + a 2 m 2 c2 - b 4 - a 2 b 2 m 2 = 0 b 2 c2 - b 2 - a 2 m 2 = 0 c2 = b2 + a 2m2 c2 = b 2 + a 2 m 2 tangent to the ellipse b 0 is a necessary and sufficient condition for the line y = mx + c to be a t x2 y2 + = 1. a2 b2 Hence, for any given value of m, the equations of tangents are y = mx b2 + a 2 m 2 . Example 1. Find the equation of the locus of a variable point P moving on the xy plane in such a way that the two tangents drawn from P to the ellipse x2 y2 + = 1 are perpendicular. a2 b2 y P _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ x Pure Maths/Ellipse/p.6 _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ 2. Find the equation of the common tangents to the ellipse x2 + 2y2 = 6 and the circle 2x2 + 2y2 = 9 . _______________________________________________ y _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ x _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ 3. Let P and Q be two points on the ellipse x2 y2 + = 1 with eccentric angles and a2 b2 respectively. Show that, if - is a constant, then the locus of the mid-point of the chord PQ is also an ellipse. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Pure Maths/Ellipse/p.7 ________________________________________________________________________ 4. If from an external point T(h, k), tangents TP and TQ are y x2 y2 drawn to the ellipse + = 1 , show that the area of a2 b2 the quadrilateral OPTQ is T(h, k) b2h 2 + a 2k 2 - a 2b2 . P Q _______________________________________________ O _______________________________________________ _______________________________________________ _______________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Note : Equation of normal at the point (x1 , y1 ) to the ellipse b2 x1 dy = d x (x , y ) a 2 y1 1 1 x2 y2 + = 1 : 2 2 a b Equation of mormal at (x1 , y1 ) : a 2 y1 y - y1 = x - x1 b2 x1 a2 x b2 y = a 2 - b2 x1 y1 If (x1 , y1 ) is expressed as (a cos , b sin ), then the equation of the normal is : x Pure Maths/Ellipse/p.8 ax cos by = a 2 - b2 sin where cos 0 and sin 0 Diameter and Conjugate Diameters y Equation of the chord with T(h, k) as the mid-point : r1 x = h + r cos y = k + r sin h + r cos 2 + a2 k + r sin r2 2 b2 = 1 For T to be the mid-point of the chord, r1 = - r2 . sum of roots = 0 2b2h cos + 2a 2k sin b2 cos2 + a 2 sin 2 2b2 h cos = - 2a 2 k sin tan = - = 0 b2 h a 2k Equation of the chord with (h, k) as mid-point : y-k =- b2 h x - h a 2k a 2k (y - k) = - b2h x - h a 2 k y + b2 h x = a 2 k 2 + b2 h 2 OR hx ky h2 k2 + = + a2 b2 a2 b2 b2 h where the slope of the chord = a 2k If m is used to denote the slope, then m = - x T(h,k) x with roots r1 and r2 2h cos 2k sin + 2 a b2 = 0 cos2 sin 2 + a2 b2 x2 y2 + = 1 2 2 a b b2 h . a 2k Pure Maths/Ellipse/p.9 ∴ k = - b2 h a 2m y = - b2 x a 2m y The mid-points of parallel chords with common slope m lie on the straight line y = - b2 x . a 2m x The mid-points of a system of parallel chords with common slope m on a line with slope - b2 and passing through the centre of a 2m the ellipse. In other words, the mid-points lie on a diameter. The diameters are said to be conjugate to one another if each of them bisects the chords parallel to the other. y = mx 2 y = - b x a 2m The product of the slopes of two conjugate diameters is - slope of OC = b tan a slope of OD = b tan a b2 . a2 y D(a cos , b sin ) C(a cos , b sin ) For CC’ and DD’ to be conjugate diameters, b2 b2 tan tan = a2 a2 As ∴ tan tan = tan ( - ) = -1 tan - tan , 1 + tan tan - = 2 If CC’ = DD’, then they are called equiconjuagte diameters. O x D’ C’ Pure Maths/Ellipse/p.10 Example 1. If OP and OQ are conjugate semi-diameters of the ellipse point P from OQ, show that OD = x2 y2 + = 1 , p is the distance of the a2 b2 ab and that the mid-point of the chord PQ lies on the ellipse p x2 y2 1 . + = 2 2 2 a b y ___________________________________________________ ___________________________________________________ ___________________________________________________ ___________________________________________________ Q P p D O x ___________________________________________________ ___________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 2.(1982) Given an ellipse (E) : x2 y2 + = 1. a2 b2 (a) P is a point (h, k). If y = mx + c is a tangent drawn from P to (E), show that a 2 - h 2 m2 + 2hkm + b2 - k 2 = 0 . (b) Two non-parallel lines of slopes m1 and m2 are equally inclined to the line y = nx. Show that 1 - n 2 m1 + m2 + 2n m1m2 -1= 0 . (c) Tangents of (E) equally inclined to the line y = nx intersect at the point P. Using the results of (a) and (b), or otherwise, find the equation of the locus of the point P. Pure Maths/Ellipse/p.11 (a) ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ (b) ________________________________________________ P.M./Ellipse/p.11 ________________________________________________ 1 - 2 y _______________________________________________ 2 ________________________________________________ ________________________________________________ ________________________________________________ 1 O 3 4 y = nx ________________________________________________ ________________________________________________________________________ (c) ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ W.42 Q. 2 , 3 , 4 H.W. W.42 Q 5 - 7 W.43 Ex.B Q. 4 , 5, 11 END x