PROBLEMS 05 - ELLIPSE Page 1
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1 PROBLEMS 0 ELLIPSE Pge 1 ( 1 ) The edpoits A d B of AB re o the X d Yis respectivel If AB > 0 > 0 d P divides AB from A i the rtio : the show tht P lies o the ellipse 1 ( ) If the feet of the perpediculrs drw to the tget t poit of the ellipse 1 from foci S d S re L d L respectivel the show tht SL S' L' ( 3 ) Prove tht the lie segmet of tget etwee the tgets t the edpoits of the mjor is forms right gle t either focus of the ellipse ( 4 ) Show tht the equtio of the chord joiig the poits P ( α ) d Q ( β ) of the ellipse α β α β α β 1 is cos si cos ( ) If the chord joiig the poits P ( α ) d Q ( β ) of the ellipse through the focus ( e 0 ) the prove tht α t β t e 1 e 1 1 psses 1 suteds right gle t the cetre the show tht t α t β 0 d if it forms right gle t the verte ( 0 ) the show tht α β t t 0 ( 6 ) If the chord joiig the poits P ( α ) d Q ( β ) of the ellipse ( 7 ) If the differece of eccetric gles of the poits P d Q o the ellipse π 1 is d PQ cuts itercepts of legth c d d o the es the prove tht c d
2 PROBLEMS 0 ELLIPSE Pge ( 8 ) If two rdii CP d CQ of the ellipse 1 re perpediculr the prove tht where C is the cetre of the ellipse CP CQ ( 9 ) Fid the equtios of the tgets drw to the ellipse from the poit ( 3 ) [ As: ] ( 10 ) Fid the equtios of the tgets of the ellipse prllel to the lie Also oti the coordites of the cotct poits As : 0 t 8 9 d 0 t 8 9 ( 11 ) Show tht the tgets t the edpoits of focl chord of the ellipse itersect o the directri ( 1 ) Show tht the poit of itersectio of the tgets to the ellipse the poits whose eccetric gles differ π lies o the ellipse 1 t ( 13 ) The differece of the eccetric gles of the poits P d Q o the ellipse π 1 is If the tgets t P d Q itersect i R the prove tht CR d PQ isect ech other ( 14 ) P d Q re coheret poits o the ellipse 1 d its uilir circle Show tht the tgets to the ellipse t P d the circle t Q itersect o the Xis ( > )
3 PROBLEMS 0 ELLIPSE Pge 3 ( 1 ) If the legths of the perpediculr liesegmets from the cetre to two mutull orthogol tgets of the ellipse 1 re p 1 d p the prove tht p 1 p ( 16 ) A tget to the ellipse 1 itersects the es i C d D respectivel d touches the ellipse t midpoit of CD i the first qudrt Fid its equtio As : ( 17 ) If the perpediculr distce of the focus S from the tget t poit P o the p ellipse 1 is p the prove tht SP p ( 18 ) B ( 0 ) is oe edpoit of the chord of ellipse 1 ( > ) pssig through the focus S If P is other edpoit the show tht the slope of CP is 3 ( 1 e ) where C is the cetre of the ellipse e ( 19 ) The foot of the perpediculr from poit P o ellipse to the mjor is is M If PM itersects the tget t the edpoit of ltus rectum i R the prove tht MR SP ( 0 ) If the lie cotiig foclchord of the ellipse circle i Q d Q the prove tht SQ SQ 1 itersects the uilir ( 1 ) Prove tht if the tget t poit P to the ellipse itersects directri t F the PF forms right gle t the correspodig focus
4 PROBLEMS 0 ELLIPSE Pge 4 ( ) The tget t poit P of ellipse itersects the mjor is i T The lie pssig AA ' itersects AP d A ' P i Q through T d perpediculr to mjor is d Q respectivel Show tht T is the midpoit of QQ ' ( 3 ) The tget t poit P of ellipse 1 itersects the es i T d T respectivel If R is the foot of perpediculr from the cetre C to the tget the prove tht TT PR ( 4 ) Fid the coditio tht the lie l m 0 m e tget to the ellipse 1 d fid the coordites of its poit of cotct As : l m l m ( ) If the tget t poit of ellipse 1 with cetre C meets the mjor is i T d mior is i T the prove tht 1 ( > ) CT CT' ( 6 ) Fid the coditio for the lie cos α si α p to e tget to the ellipse 1 [ As: p cosec α cot α ] ( 7 ) P d Q re correspodig poits o ellipse d its uilir circle respectivel If the tget t P to the ellipse meets the mjor is i T the show tht QT is tget to the uilir circle ( 8 ) Fid the perpediculr distce etwee the tgets to the ellipse which re prllel to the lie [ As: 4 / ]
5 PROBLEMS 0 ELLIPSE Pge 1 which hs its mid ( 9 ) Prove tht the equtio of the chord of the ellipse h h poit t ( h ) is ( 30 ) Prove tht the equtio of the chord joiig the poits P ( α β ) d Q ( α β ) o the ellipse 1 is cos α si α cos β ( 31 ) Prove tht the re of the trigle formed the poits P ( θ ) Q ( α ) d R ( β ) o the ellipse 1 is α β si β si θ θ α si ( 3 ) Prove tht the equtios of the commo tgets to the ellipse 1 d the circle r ( < r < ) re r ± r ± r ( 33 ) Circles of costt rdius c re drw to pss through the eds of vrile dimeter of the ellipse Prove tht the locus of their cetres is he curve ( ) ( ) c ( ) ( 34 ) Prove tht the mesure of the gle etwee the two tgets drw to the ellipse 1 from eterl poit ( h ) is t 1 h h 1
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