Drill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100.

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1 Drill Exercise 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 = 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 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 x 5 The foci of hperol coincide with the foci of the ellipse = 1, then find the eqution of the hperol if its eccentricit is Drill Exercise Find the equtions of the smptotes of the hperol, 3x + 10x x = 0 Find the eqution of the conjugte hperol of the hperol, 3x 5x + 5x = 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

2 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 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 = 0 5 If rectngulr hperol circumscries tringle, prove tht it lso psses through its orthocentre Drill Exercise 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 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 =

3 Drill Exercise 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 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 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 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

4 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 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

5 Answer Ke Drill Exercise e, ( 5,0),( 9,0), x (- 1, 0) Drill Exercise x = 15 pir of rs 3 x = 0 4 x = sec, = tn Drill Exercise x = 0 nd x = 0 3x 5x + 5x = 0 3 6x + 13x x = 0 Drill Exercise (LR) = (TA) = (CA)= 1 ; e = ; foci ( 6,6 ), ( 6, 6 ) ; vertex = (6, 6), ( 6, 6); eqution of directrices = x + = ± 6 0 Drill Exercise point ( 7, 3) lies inside ; (, 7) lies outside x = 3x + 5, = 3x 5 4 4, Drill Exercise = 3x m6 - m = 0 Drill Exercise = e Drill Exercise x 16 = m 1 + m = 4/11, m 1 m = 0/11 Drill Exercise (x + ) + 4 (x ) = x x 38x =

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2 SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the