hsn.uk.net Page 1 Circle Find the equation of the tangent at the point (3, 4) on the circle x 2 +y 2 +2x 4y 15 =0. 4 Higher Mathematics
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1 Circle 1. Find the equation of the tangent at the point (3, 4) on the circle x 2 +y 2 +2x 4y 15 = C CN G2,G5,G9 1996P1Q4 hsn.uk.net Page 1
2 2. (a) Find the equation of AB, the perpendicular bisector of the line y Q(1, 9) joing the points P( 3,1) and Q(1,9). A 4 (b)cisthecentreofacirclepassing throughpandq.giventhatqcis parallel to the y-axis, determine the C equation of the circle. 3 (c)thetangentsatpandqintersectat T. Write down (i)theequationofthetangentatq P( 3, 1) (ii) the coordinates of T. 2 O B x (a) 4 C CN G7 x +2y =9 2000P2Q2 (b) 3 C CN G10 (x 1) 2 + (y 4) 2 =25 (c) 2 C CN G11,G8 (i)y =9,(ii)T( 9,9) 1 ss: knowtousemidpoint 2 pd: processgradientofpq 3 ss: knowhowtofindperp.gradient 4 ic: stateequ.ofline 5 ic: interpret paralleltoy-axis 6 pd: processradius 7 ic: stateequ.ofcircle 8 ic: interpretdiagram 9 ss: knowtouseequ.ofab 1 midpoint = ( 1,5) 2 m PQ = ( 1) 3 m = y 5 = 1 2 (x ( 1)) 5 y C =4 statedorimpliedby 7 6 radius =5orequiv. statedorimpliedby 7 7 (x 1) 2 + (y 4) 2 =25 8 y =9 9 T= ( 9,9) hsn.uk.net Page 2
3 3. 2 C CN G P1Q4 4. FindtheequationofthecirclewhichhasP( 2, 1)andQ(4,5)astheendpoints of a diameter. 3 3 C CN G P1Q9 5. Theliney = 1isatangenttoacirclewhichpassesthrough (0,0)and (6,0). Find the equation of this circle. 6 1 C CN G P1 Q20 5 A/B CN G9,G15 hsn.uk.net Page 3
4 6. (a) 4 C CN G P2 Q8 (b) 4 C CN G10 hsn.uk.net Page 4
5 7. (a) 3 C CN G2,G3 1992P2Q9 (b) 1 C CN G10 (c) 2 C CN G12, G13 (c) 3 A/B CN G12, G13 hsn.uk.net Page 5
6 8. (a) 3 C CN G5,G3 1993P2Q3 (b) 5 C CN G10 hsn.uk.net Page 6
7 9. (a) 4 C CN G5,G3 1991P2Q2 (b) 6 C CN G10, G1 hsn.uk.net Page 7
8 10. CirclePhasequationx 2 +y 2 8x 10y +9 =0. CircleQhascentre ( 2, 1) andradius2 2. (a) (i)showthattheradiusofcirclepis4 2. (ii)henceshowthatcirclespandqtouch. 4 (b)findtheequationofthetangenttothecircleqatthepoint ( 4,1). 3 (c)thetangentin(b)intersectscirclepintwopoints.findthex-coordinatesof thepointsofintersection,expressingyouanswersintheforma ±b 3. 3 (a) 2 C CN G9 proof 2001 P1 Q11 (a) 2 A/B CN G14 (b) 3 C CN G11 y =x +5 (c) 3 C CN G12 x =2 ±2 3 1 ic: interpretcentreofcircle(p) 2 ss: findradiusofcircle(p) 3 ss: findsumofradii 4 pd: comparewithdistancebetween centres 5 ss: findgradientofradius 6 ss: usem 1 m 2 = 1 7 ic: stateequationoftangent 8 ss: substitutelinearintocircle 9 pd: expressinstandardform 10 pd: solve(quadratic)equation 1 C P = (4,5) 2 r P = = 32 =4 2 3 r P +r Q = =6 2 4 C P C Q = = 6 2and so touch 5 m r = 1 6 m tgt = +1 7 y 1 =1(x +4) 8 x 2 +(x+5) 2 8x 10(x+5)+9 =0 9 2x 2 8x 16 =0 10 x =2 ±2 3 hsn.uk.net Page 8
9 11. (a) 5 C CN G P2 Q1 (bi) 1 C CN G9 (bii) 3 C CN G8, G3 hsn.uk.net Page 9
10 12. (a) (i)showthatthelinewithequationy=3 xisatangenttothecirclewith equationx 2 +y 2 +14x +4y 19 =0. (ii) Find the coordinates of the points of contact, P. 5 (b)relativetoasuitablesetofcoordinateaxes,thediagrambelowshowsthe circlefrom(a)andasecondsmallercirclewithcentrec. P C Theliney =3 xisacommontangentatthepointp. Theradiusofthelargercircleisthreetimestheradiusofthesmallercircle. Find the equation of the smaller circle. 6 (ai) 4 C CN G13 proof 2010 P2 Q3 (aii) 1 C CN G12 P( 1, 4) (b) 6 B CN G9,G15 (x 1) 2 + (y 6) 2 =8 1 ss: substitute 2 pd: expressinstandardform 3 ic: startproof 4 ic: completeproof 5 pd: coordinatesofp 6 ic: statecentreoflargercircle 7 ss: findradiusoflargercircle 8 pd: findradiusofsmallercircle 9 ss: strategyforfindingcentre 10 ic: interpretcentreofsmallercircle 11 ic: stateequation 1 x 2 +(3 x) 2 +14x+4(3 x) 19 =0 2 2x 2 +4x +2 =0 3 2(x +1)(x +1) 4 equalrootssolineisatangent 5 x = 1,y =4 6 ( 7, 2) e.g. Steppingout 10 (1,6) 11 (x 1) 2 + (y 6) 2 =8 hsn.uk.net Page 10
11 13. Findthepossiblevaluesofkforwhichthelinex y =kisatangenttothecircle x 2 +y 2 = C CN A18,A P1Q18 3 A/B CN G Explainwhytheequationx 2 +y 2 +2x +3y +5 =0doesnotrepresentacircle. 2 2 C CN G9 1993P1Q Forwhatrangeofvaluesofcdoestheequationx 2 +y 2 6x +4y+c =0represent acircle? 3 2 C CN G9 1997P1Q14 1 A/B CN G9 hsn.uk.net Page 11
12 16. (a) 3 C CN G9,G5,C4 1996P2Q10 hsn.uk.net (a) 3 A/B CN G9,G5,C4 Page 12 (b) 4 A/B CN G11,G10,G15
13 17. Forwhatrangeofvaluesofkdoestheequationx 2 +y 2 +4kx 2ky k 2 =0 represent a circle? 5 5 A NC G9,A17 forallk 2000P1Q6 1 ss: knowtoexamineradius 2 pd: process 3 pd: process 4 ic: interpretquadraticinequation 5 ic: interpretquadraticinequation 1 g =2k,f= k,c = k 2 statedorimpliedby 2 2 r 2 =5k 2 +k+2 3 (realr )5k 2 +k+2 >0(accept ) 4 usediscr.orcompletesq.ordiff. 5 trueforallk hsn.uk.net Page 13
14 18. Part Marks Level Calc. Content Answer U3 OC1 (a) 3 C CN G P2 Q6 (b) 3 C CN G9, G25 (c) 3 A/B CN CGD hsn.uk.net Page 14
15 19. Part Marks Level Calc. Content Answer U3 OC1 8 C CN G9,G10,G P2Q8 [END OF QUESTIONS] hsn.uk.net Page 15
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