FP3 past questions - conics

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1 Hperolic functions cosh sinh = sinh = sinh cosh cosh = cosh + sinh rcosh = ln{ + } ( ) rsinh = ln{ + + } + rtnh = ln ( < ) FP3 pst questions - conics Conics Ellipse Prol Hperol Rectngulr Hperol Stndrd Form + = = 4 = = c Prmetric Form ( cosθ, sinθ ) ( t, t) ( sec θ, tn θ ) (± cosh θ, sinh θ ) ct, c t Eccentricit e < = ( e ) e = e > = e ( ) e = Foci ( ± e, 0) (, 0) ( ± e, 0) (± c, ± c) Directrices = ± = e = ± + = ± c e Asmptotes none none = ± = 0, = 0 Edecel AS/A level Mthemtics Formule List: Further Pure Mthemtics FP3 Issue Septemer 009

2 Differentition f() f () rcsin rccos rctn + sinh cosh cosh sinh tnh sech rsinh rcosh + rtnh Integrtion (+ constnt; > 0 where relevnt) f() f( ) d sinh cosh cosh sinh tnh ln cosh + + rcsin rctn ( < ) rcosh, ln{ + } rsinh, ln + ln + ln { + + } = rtnh ( ( > ) < ) Edecel AS/A level Mthemtics Formule List: Further Pure Mthemtics FP3 Issue Septemer 009

3 Core Mthemtics C4 Cndidtes sitting C4 m lso require those formule listed under Core Mthemtics C, C nd C3. Integrtion (+ constnt) f() f( ) d sec k tn k k tn ln sec cot ln sin cosec ln cosec + cot, ln tn( ) sec ln sec + tn, ln tn( + 4 π ) dv du u d = uv v d d d Edecel AS/A level Mthemtics Formule List: Core Mthemtics C4 Issue Septemer 009 7

4 FP3 conics pst-pper questions 6. The hperol H hs eqution, where nd re constnts. The line L hs eqution m c, where m nd c re constnts. () Given tht L nd H meet, show tht the -coordintes of the points of intersection re the roots of the eqution ( m ) mc( c ) 0 () Hence, given tht L is tngent to H, () show tht m c. The hperol H hs eqution. 5 6 (c) Find the equtions of the tngents to H which pss through the point (, 4). Lines re = + 3 nd = 6 () (7) 6 *M3545A068*

5 . The line = 8 is directri of the ellipse with eqution + =, > 0, >0, nd the point (, 0) is the corresponding focus. Find the vlue of nd the vlue of. (5) = 4 = sqrt(3) *N35389RA08*

6 8. The hperol hs eqution =. 6 4 The line l is the tngent to t the point (4sec, tn). () Use clculus to show tht n eqution of l is sint = 4cost (5) The line l psses through the origin nd is perpendiculr to l. The lines l nd l intersect t the point. () Show tht, s vries, n eqution of the locus of is ( ) = (8) 4 *N35389RA048*

7 8. The hperol H hs eqution = () Use clculus to show tht the eqution of the tngent to H t the point ( cosh θ, sinh θ ) m e written in the form coshθ sinhθ = (4) The line l is the tngent to H t the point ( cosh θ, sinh θ ), θ. Given tht l meets the -is t the point P, () find, in terms of nd, the coordintes of P. () The line l is the tngent to H t the point (, 0). Given tht l nd l meet t the point Q, (c) find, in terms of, nd, the coordintes of Q. () (d) Show tht, s vries, the locus of the mid-point of PQ hs eqution (4 + ) = P is (/ cosh thet,0) Q is (, [cosh thet - ] / sinh thet) (6) 4 *P3544A048*

8 . The hperol H hs eqution 6 9 = Find () the coordintes of the foci of H, () the equtions of the directrices of H. (3) () e = 5/4 Foci (±5,0), Directrices = ±6/5 *P40A03*

9 6. The ellipse E hs eqution + = The line l is tngent to E t the point P ( cos θ, sin θ). () Using clculus, show tht n eqution for l is cosθ sinθ + = (4) The circle C hs eqution + = The line l is tngent to C t the point Q ( cos θ, sin θ). () Find n eqution for the line l. () Given tht l nd l meet t the point R, (c) find, in terms of, nd, the coordintes of R. (3) (d) Find the locus of R, s vries. () 8 *P40A083*

10 . The hperol H hs foci t (5, 0) nd ( 5, 0) nd directrices with equtions 9 9 = nd =. 5 5 Find crtesin eqution for H. (7) *P4956A03*

11 3. The point P lies on the ellipse E with eqution + = 36 9 N is the foot of the perpendiculr from point P to the line = 8 M is the midpoint of PN. () Sketch the grph of the ellipse E, showing lso the line = 8 nd possile position for the line PN. () () Find n eqution of the locus of M s P moves round the ellipse. (4) (c) Show tht this locus is circle nd stte its centre nd rdius. (3) 8 *P4956A083*

12 . A hperol H hs eqution =, where is positive constnt. 5 The foci of H re t the points with coordintes (3, 0) nd ( 3, 0). Find () the vlue of the constnt, (3) () the equtions of the directrices of H. (3) =, directri = plus or minus 44/ 3 *P4343A08*

13 7. The ellipse E hs eqution + =, > > 0 The line l is norml to E t point P ( cos θ, sin θ), 0 < θ< () Using clculus, show tht n eqution for l is π sin cos = ( )sincos (5) The line l meets the -is t A nd the -is t B. () Show tht the re of the tringle OAB, where O is the origin, m e written s ksin, giving the vlue of the constnt k in terms of nd. (4) (c) Find, in terms of nd, the ect coordintes of the point P, for which the re of the tringle OAB is mimum. (3) P is / sqrt (), / sqrt () 0 *P4343A008*

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