Drill Exercise - 1 1 Find the coordintes of the vertices, foci, eccentricit nd the equtions of the directrix of the hperol 4x 5 = 100 Find the eccentricit of the hperol whose ltus-rectum is 8 nd conjugte xis is equl to hlf the distnce etween the foci 3 Find the coordintes of the centre of the hperol, x + 3x + + x + 3 + = 0 4 Find the eccentricit of the hperol = 1 which psses through (3, 0) nd (3, ) x 1 5 The foci of the ellipse + 16 = 1 nd the hperol coincide, then find the vlue 144 81 5 of Drill Exercise - 1 Find the eqution of the set of ll points such tht the difference of their distnces from (4, 0) nd ( 4, 0) is lws equl to Find the locus of the point which stisfies (x 5) (x 5) 10 3 Find the eqution of hperol with coordinte xes s principl xes nd the distnces of one of its vertices from the foci re 3 & 1 (x 5) ( 3) 4 Find the prmetric eqution of the hperol 1 36 5 x 5 The foci of hperol coincide with the foci of the ellipse + 5 9 = 1, then find the eqution of the hperol if its eccentricit is Drill Exercise - 3 1 Find the equtions of the smptotes of the hperol, 3x + 10x + 8 + 14x + + 7 = 0 Find the eqution of the conjugte hperol of the hperol, 3x 5x + 5x + 11 8 = 0 3 The smptotes of the hperol re prllel to x + 3 = 0 nd 3x + = 0 Its centre is (1, ) nd its psses through (5, 3) Find the eqution of hperol http://kj59wordpresscom
4 The ordinte of n point P on the hperol 5x 16 = 400 is produced to cut its smptotes in points Q nd R Prove tht QPPR = 5 5 Prove tht the product of the perpendiculrs from n point on the hperol smptotes is equl to x 1 to its Drill Exercise - 4 1 Find the lengths of trnsverse nd conjugte xes, eccentricit nd coordinte of foci nd vertices, length of the ltusrectum, eqution of the directrices of the rectngulr hperol x = 36 The distnce etween the directrices of rectngulr hperol is 10 units, then find the distnce etween its foci 3 If circle cuts rectngulr hperol x = c in A, B, C, D nd the prmeters of these four points e t 1, t, t 3 nd t 4 respectivel Then show tht t 1 = t 4 If the tngent nd norml to rectngulr hperol cut off intercepts 1 nd on one xis nd 1 nd on the other show tht 1 + 1 = 0 5 If rectngulr hperol circumscries tringle, prove tht it lso psses through its orthocentre Drill Exercise - 5 1 Find the positions of the points (7, 3) nd (, 7) reltive to the hperol 9x 4 = 36 Find the eqution of the tngent to the hperol x 4 = 36 which is perpendiculr to the line x + 4 = 0 3 Find the eqution of the tngent to the hperol x 3 = 6 which is prllel to the line = 3x + 4 4 Find the point of contct of the line = x 1 with hperol 3x 4 = 1 x 5 Find the vlue of m for which = mx + 6 is tngent to the hperol = 1 100 49 http://kj59wordpresscom
Drill Exercise - 6 1 If the tngent t the point (h, k) to the hperol x / / = 1 cuts the uxilir circle in points whose ordintes re 1 nd then prove tht 1/ 1 + 1/ = /k Show tht the re of the tringle formed the lines x = 0, x + = 0 nd n tngent to the hperol x = is 3 The tngent t n ritrr point P on the locus of mid point of PQ x = 1 meets the line x = 0 t point Q, then find 4 Find the eqution of the tngent to the hperol x - 3 = 6 which is prllel to the line = 3x + 4 5 Find the conditions tht stright line with slope m will e norml to prol = 4x s well s tngents to rectngulr hperol x - = Drill Exercise - 7 x 1 Find the eqution of norml to the hperol = 1 t ( 4, 0) 16 9 The norml to the hperol x 1 drwn t n extremit of its ltus rectum is prllel to n d1 5i smptote Show tht the eccentricit is equl to the squre root of 3 If the tngent nd the norml to the rectngulr hperol x = c, t point, cuts off intercepts 1 nd on the x-xis nd 1, on the -xis, then find the vlue of 1 + 1 4 The norml t P to hperol of eccentricit e, intersects its trnsverse nd conjugte xes t L nd M respectivel If locus of the mid point of LM is hperol, then find the eccentricit of the hperol 5 If the norml t P to the hperol =1 meets the trnsverse xis in G nd conjugte xis in g nd CF e perpendiculr to the norml, from the centre then prove tht PFPG = CB =, PF Pg = CA = Also prove tht SG = e SP (where S is the focus) Drill Exercise - 8 1 Find the numer of point(s) outside the hperol cn e drwn to the hperol 1 from where two perpendiculr tngents 5 36 Find the eqution to the chords of the hperol, 5x 16 = 400 which is isected t the http://kj59wordpresscom
point (6, ) 3 If m 1 nd m re the slopes of the tngents to the hperol x 1 which pss through the 5 16 point (6, ) then find the vlue of m 1 m nd m 1 + m 4 If the chord through the points ( sec, tn ) nd ( sec, tn ) on the hperol = 1 1 e, psses through focus, then prove tht tn tn = 1 e 1 e, 1 e for for focus(e,0) focus( e,0) 5 Tngents re drwn from n point on the hperol x = + to the hperol = 1, then prove tht the meet the xes in concclic points Drill Exercise - 9 1 The chord of the hperol x / / = 1 whose eqution is x cos + sin = p sutends right ngle t the centre Prove tht it lws touches circle Find the product of the length of the perpendiculrs drwn from foci on n tngent to the hperol (x / ) - ( / ) = 1 3 Find the locus of the point, tngents from which to the rectngulr hperol x = contin n ngle of 45º 4 Show tht the norml to the rectngulr hperol x = c t the point t meets the curve gin t the point t 1 such tht t 1 t 3 = 1 5 The smptotes of hperol re prllel to lines x + 3 = 0 nd 3x + = 0 The hperol hs its centre t (1, ) nd it psses through (5, 3) Find its eqution http://kj59wordpresscom
Answer Ke Drill Exercise - 1 1 9 5 e, ( 5,0),( 9,0), x 5 9 4 3 3 (- 1, 0) 4 13 3 5 7 Drill Exercise - 1 15x = 15 pir of rs 3 x 3 + 3 = 0 4 x = 5 + 6 sec, = 3 + 5 tn 5 1 4 1 Drill Exercise - 3 1 3x + 4 + 5 = 0 nd x + + 3 = 0 3x 5x + 5x + 11 16 = 0 3 6x + 13x + 6 38x 37 98 = 0 Drill Exercise - 4 1 (LR) = (TA) = (CA)= 1 ; e = ; foci ( 6,6 ), ( 6, 6 ) ; vertex = (6, 6), ( 6, 6); eqution of directrices = x + = ± 6 0 Drill Exercise - 5 1 point ( 7, 3) lies inside ; (, 7) lies outside x 3 3 0 3 = 3x + 5, = 3x 5 4 4, 3 5 3 Drill Exercise - 6 3 4 4 = 3x + 5 5 m6 - m = 0 Drill Exercise - 7 1 = 0 3 0 4 e Drill Exercise - 8 1 0 75x 16 = 418 3 m 1 + m = 4/11, m 1 m = 0/11 Drill Exercise - 9 1 3 (x + ) + 4 (x ) = 4 4 5 6x + 6 + 13x 38x 37 98 = 0 17 0 http://kj59wordpresscom