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Chapter - Ellipse class 11th, Study notes of Mathematics

Mathematics

Complete summary and explanation of chapters with example providing full knowledge from Basics to advance for competitive exams

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2023/2024

Available from 07/09/2024

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: B:Ellipse . gandard Equation and pefinitlan ; wation ofan ellipse gai yordinate axes is we 5 2 referred to its Principal axes vet a>band R=a(l-e) ; a-Paaee e =eccentricity (0 SP + SP = 2a= Major axis. « The sum of the focal distances of any point on the ellipse is equal to the Thajor axis. Hence, dis- tance of focus from the extremity of a minor axis is equal to semi major axis, ie. BS Cal, ~ Tf the equation of the ellipse is given as and nothing is mentioned, then th assume that a > d, e rule is to neem PCT TTT peta re, Chapter 14 If LR of an ellipse i eccentricity is Ipse is half of its minor axis, then its SOLUTION 2 As given 2b" b rs 2b a > 4B? gt = 4a?(1—e) a? > 1-2 14 ce: e= V3/2. exam Findtheequationoftheellipsewhosefociare(2,3),(-2,3) and whose semi minor axis is of length V5. SOLUTION Here Sis (2, 3) and S” is (-2, 3) and b= V5 => SS =4=2ae => ae=2 but B=a(l-¢’) => 5S=a-4>a=3. Hence, the equation to major axis is y= 3 Centre of ellipse is midpoint of SS” ie, (0, 3) . Equation to ellipse is 2 _ 42 2% Oo 2) 2. 2 _ 32 or #,o 3) =1 9 5 esta Find the equation of the ellipse having centre at (1, 2) one focus at (6, 2) and passing through the point (4, 6). SOLUTION th centre at (1, 2) the equation of the ellipse is Wil 2 _ 22 (x- I) 2 = 1. a b It passes thro gh {he point (4, 6) : 9 , 16 1 => at P a | pistance petween the focus and the centre it (6-) 5 ae ays gave a Solving for g@and 8 we get a 45 B20. lipse is ~25 nN from the Equations (1) ai | and. Hence, the equation of the el 2 _ 2 oe? = 45 20 A 2 2 XR aS Spee) _ AA’ = Minor axis = 2a | BBY = Major axis = 2b @=b(1-e’) . Latus rectum Beppe ’ 2a? : LU =L Ly = = equation y=xtbe Y A Directrix 2 v Y et} \ Chapter 14 near The line = line y= mx +c meets the ellipse an 1 in two Oints re Poe . a by . points real, coincident or imaginary according to as 18 | <=or> anes b. | Hence, y= mx +c is tangent to the ellipse | eye } a bt The equation to the chord of the ellipse joining with eccentric angles o and B is given by +B yo, 42 sin = = Life=atn?s b*. two points x at = cos B a The equations x = cos Qandy=bsin® together represent the ellipse where 6 is a parameter (eccentric angle). s @, b sin @) is on the ellipse then; If P (8) = (4 ©o! on the auxiliary circle. Q (8) =(acos 8,4 sin 6) is , - Ee 2 For what value of % does the line y= 3+ A touches the ellipse ox2-+ 16y?= 144. SOLUTION +; Equation of ellipse is 2 2 voy 9x2 + 16? = 144 or —— + = iy ato os Comparing this with xy ae + ep = | then we get a?= 16 and P=9 and comparing the line ext hwithy ame te mal and c= ye eth touches the ellipse ine ¥ =" jf the 4 167= 144, then =a? m4. pp 42 = 16% +9 = q2=25 a pets. ‘3 J B are eccentric angles of end points of a; If a, : pica ee hielo tin ; chord of the ellips? “7 + ay then tan 0/2 - ti, js equal to SOLUTION : Equation of line joining points “a” and ‘B’ is x atB YY a+B =) + —sin— = 008 ee te = cos S= 6 2 If it is a focal chord, then it passes through i (ae, 0), 80 a= ecos eos 2 cos aaB F ae cos out: B 1 2 cos t= B — ggg % +8 > 2 a cos 2 = Big ea ont B e+l 3 2 7 2sino/2 sinB2 _ 2— =o 2eosa/2 cosp/2. e+! > - mane tan 2 =f Qhs2 ett ar ie —_ 2 ;, Pair of tangents are drawn to an ellipse 74 y ba 2 : xy | 8 [fS, H are the foci of an ellipse +4 = land a dire bins of the point of intersection aright angtes is called the Di qs gy Cthlre of the ellipse and whose r\ Find the equation of the lay centre (1,0) that can be ingeri pt ayr= 16. +, Find the condition so that the line px + Wy reest circle with bed in the ellipse x y’ =r inter- sects the ellipse —-+=-_ =|, . sects Ip ae Ee 1 in points whose eccen-tric angles differ by ce 4 a Bb 1 (a> 4) from the foot of the directrics. Find the co-ordinates of the point of intersection of these tangents with the minor axis. 4, Ifp and p’ be the perpendiculars from the center 2 = 2 +. x and a focus of the ellipse + = I on any tan- gent and r the focal distance of the point of contact, then prove that p’a =p r. A is any point on the curve. ASB, BHC, CSD, DHE,...... are chords and 6, 8,, 8,, 8, are the eccentric angles of A, B, C, D, respectively, then prove that, 6 8. 6. 8, 8. tan tan 2 cot 2 -cot 8 = tan tan =e 2 2 2 2 2 2 6. A line is drawn parallel to the minor axis of an 2 V2 ellipse aa = 1, mid way between a focus a and corresponding directrix; Prove that the product of the perpendiculars on it from the extremities of any chord with eccentric angle 9, and 6, passing through the corresponding focus is independent of 8, and @,. of the tangents which irector circle. The equa- pei.e.,a circle whose centre dius is the length of ae, A minor axis. ‘ne joining the ends of the major and lothis locus is x?+y?=a?+ Parabola, Ellipse, and Hyperbola EEN A tangent to the ellipse 2+ 4y?= 4 meets the ellipse 4+2=6 at Pand Q. Prove that the tangents at Pand Q of the ellipse x? + 2y°= 6 are at right angles. SOLUTION Given ellipse are 2 2 il a) 4 I 2 2 and 4% 21 (2) 6 3 any tangent to Equation (1) is xcos@ | ysin® _ | (3) 2 1 It cuts Equation (2) at P and Q, and suppose tangent at P and Q meet at (4, &). Then equation of chord of contact of (/, &) with respect to ellipse (2) is dey ==! 6 3 comparing Equation (3) and (4), we get (4) cos@ _ sin® Al3 Af3 > cos 8 _ 3 1 k and sin 6 _ > P+R=9 locus of the point (A, 4) is P+yP =I + P=G43 =0+b ie. director circle of second ellipse. Hence, the tangents are at right angles. 4 eiaiadasiiad 1. Point form: Equation of tangent to the its point (x,, y,) is given ellipse at year t Ch oe +b pn? metric farm: Equation of the norm, point (a cos 0, b sin 8) is al to the given * ipso at the ay + see 8 = by cosee 0 = (a?~ 22) ay be a . V garmal to the ellipse 2 + 7 =1, ggLUTION tion of normal to the given ellipse at (a cos a pain) is ax by cos@ sin@ aah 109) yihe line Le + ary =n is also normal to the ellipse then ere must be a value of @ for which line Equation (1) piline Ix + my = 7 are identical. For the value of @ we have 1 m _ n a) _ a) cos 8 sin @ 38 =— @) a l(a" —b’) =bn : (4) im(a* — b*) Sqzring and adding Equation (3) and (4), we get an Ce =a pp Fe ich is the required condition. ad sin 8 = — ipse "the normal at an end of a latus-reclum of an cllip: ‘a ne extremity of the e 55 ugh 01 ep = 1 passes throug! “nor axis, show that V5 - 1, 2 he eccentricity of the ellipse 1s ben bye = Parabola, Ellipse, and Hyperbola cae SOLUTION The co-ordinates of an end of the latus-rectum arc f (ae, Ha), The equation of normal at P (ae, bYa) is ay YO) _ ap ’ ae ba | or -_ -ay=a-b é It passes through one extremity of the minor axis whose co-ordinates are (0, -b) O+ab=r- ad (a? b*) = (a — BY => aa(l-e)=(aey? => l-@+e =ate-l =0 => (e+e-1=0 P and 2 are corresponding points on the ellipse a The normal at P to the ellipse meets CQ in R, where C is the centre of the ellipse, prove that CR=a + b. mo | =, ve 1 and the auxiliary circles, respectively. | SOLUTION Let P=(acos @, b sin 6) Q= (a cos 6, a sin 6) Equation of normal at P is (a sec 0) x — (b cosec @) » =e-h w equation of CQ is ystan Ox Q) Chapter 14 Solving Equation (1) and (2), we get (a-b) x =(a— 6") cos 8 oi x =(a+b)cos 8 and y=(a+b) sin ® oa R= ((a +b) cos 6, (a+ 6) sin 0) oe CR=atb. If P (x,, y,) be any point lies outside the ellipse 2 2 24521 a &b and a pair of tangents FA, PB can Then the equation of pair of tangents 0 S5,=T? be drawn to it from P. f PA and PB is where If PA and PB be the tangents from point P (x, y,) to the 2 x nse oe to Hl. ellipse at ge Then the equation of the chord of contact AB is 4 YA 21 Ae = ab or Ta 0 (al XH) x 7” pF a . : intersection of tan, SOLUTION P j i f intersecti Let PoC hy be the point oft on OF tang, atA and i eee of chord of contact AB is xh YR oy a +a fy he parabola. which touches t Equation of tangent to parabola = isy= + Vig eee men a => mx y= i 0 Equation (1) and (2) as must be same a com 2 See Tn # A ke 1 ae B ton m= af Le and = 7 ka? rs kat 4 = locusofPis yy=—-—'%: a AP i pruation of Chord with | (Xu. Ya deve The i Xx equation of the chord of the ellipse “7? a whose mid-poj point be (x, y,) is T= 8, ant ETA casters 8. Perpendi Tpendiculars from the centre upon all chords which Join th ; eli © ends of any perpendicular diameters of the 5 trike: Are of constant length. the . fangent at the point P of a standard cllipse meets f axes in T and ¢ and CY is the perpendicular on it ‘orm the centre then, Q@) Tt. P¥=a@— band (b) least value of 7t is a +b. 1, Let ABC be an equilateral triangle inscribed in the circle x2 + 37 = a®. Suppose perpendiculars from A, 2 2 B, C to the major axis of the ellipse, = + a =i, a so that P, Q, R lie on the same side of the major axis as A, B, C, respectively. Prove that the normals to the ellipse drawn at the points P,Q, and R are concurrent. 2. Let C, and C, be two circles with C, lying inside C). A circle C lying inside C, touches C, internally and (a > b) meet the ellipse, respectively, at P,O,R | q \ C, externally. Identify the locus of the centre of C. : i the perpend in an ellipse, : ar f 3. Prove that, tangent and the line joining rh C yg Upon A , focus ie ellipse to the point of contact meg, in, et nding directrix. an he «eA jg climinated from . tif @ is elimina’ the Feat =| and.xsin 8—ycos @=(,2 correspo! 4, Show that ZX eoso+7sin? 2 - 12, then the eleminant represent, n ellipse. Find the locus of the Dein f the pair of perpendicular tangensy Sin a + 6} cos equation ofa intersection © Jlipse- : 5 show that the equation of the tangents to the cling x + yey at the points of intersection wit, tk ath ele : tine, px tay + HOU east ge) Oats ge-N=ertayt 1). 6. Let C is the center, BCB’ the minor axis and 5 ip % x’ _ focus (a @, 0) of the ellipse, ate =1 BSis he. ellipse again in P, show that cp duced to meet the ; ith the x-axis where, 2e tang = makes an angle > wi