2.4 Circles Paper1SectionA Each correct answer in this section is worth two marks. 1.Thepoint (2, 3)liesonthecircle with equation x 2 +y 2 +6x 2y +c =0. Whatisthevalueofc? A. 31 B. 13 C. 1 3. The point P(2, 3) lies on the circle with centre C as shown.the gradient of CP is 2.What is the equation of thetangentatp? C y O P(2, 3) x D. 9 A. y +3 = 2(x 2) B. y 3 = 2(x +2) C. y +3 = 1 2 (x 2) D. y 3 = 1 2 (x +2) 2.Acirclehascentre (2,4)andpasses through ( 1, 1). Whatistheequationofthecircle? A. (x 2) 2 + (y 4) 2 = 18 4.ThepointP( 2,4)liesonthecircle with equation x 2 +y 2 2x +2y 32 =0. Whatisthegradientofthetangent tothecircleatp? A. 1 3 B. (x 2) 2 + (y 4) 2 =18 C. (x +2) 2 + (y +4) 2 =18 D. (x +2) 2 + (y +4) 2 =26 hsn.uk.net Page 1 B. 3 5 C. 1 D. 3 Questions marked c SQA
5.Acirclehasequation (x +1) 2 + (y 2) 2 =29. Whatisthegradientofthetangent tothecircleatthepoint (1, 3)? A. 2 5 B. 0 C. 5 2 D. 1 2 8.Acirclehasequation x 2 +y 2 2x 4y +1 =0. Here are two statements about the circle: I.Thecirclehascentre ( 2, 4). II.Thecirclehasradius1. Which of the following is true? A. neither statement is correct B. only statement I is correct C. only statement II is correct D. both statements are correct 6.Thelinewithequationy =2x intersects the circle with equation x 2 +y 2 =5atthepointsJandK. Whatarethex-coordinatesofJand K? A. x J =1,x K = 1 B. x J =2,x K = 2 C. x J =1,x K = 2 D. x J = 1,x K =2 7.Acirclehasequation x 2 +y 2 +8x +6y 75 =0. Whatistheradiusofthecircle? A. 5 B. 10 9.Acirclehasequation x 2 +y 2 4x +6y +4 =0. Here are two statements about the circle: I.Thecirclehascentre ( 2,3). II.Thecirclehasradius3units. Which of the following is true? A. neither statement is correct B. only statement I is correct C. 75 C. only statement II is correct D. 175 hsn.uk.net Page 2 D. both statements are correct Questions marked c SQA
10. A circle has equation x 2 +y 2 ax +2by +c =0.The centreofthecircleis ( 1,4). Whatarethevaluesofaandb? a b A. 2 4 B. 1 2 C. 2 4 D. 2 4 11. A circle has equation (x 3) 2 + (y +4) 2 =20. Findthegradientofthetangentto thecircleatthepoint (1,0). A. 2 B. 1 2 C. 1 2 13.Theequationofthecircleshownin the diagram is x 2 +y 2 6x 10y +9 =0.The x-axisandthelinelareparallel tangents to the circle. y O Whatistheequationoflinel? A. y =5 B. y =10 C. y =18 D. y =20 l x D. 2 12.Acirclehascentre (2, 1),andhas they-axisasatangent. Whatistheequationofthecircle? A. (x +2) 2 + (y 1) 2 =4 B. (x 2) 2 + (y +1) 2 =4 C. (x +2) 2 + (y 1) 2 =1 D. (x 2) 2 + (y +1) 2 =1 14.Whatisthelargestrangeofvalues ofkforwhichtheequation x 2 +y 2 6x +4y +k =0represents acircle? A. k <52 B. k <13 C. k > 13 D. Allrealk hsn.uk.net Page 3 Questions marked c SQA
[ENDOFPAPER1SECTIONA] Paper1SectionB 15. The diagram below shows the graph of the cubic with equation y =x 3 3x 2 +5x +4andacirclewith centre C. AtthepointPthelinelisatangentto boththecurveandthecircle. (a)thetangentline l hasgradient 2. Find the coordinates of P. Q 5 (b) The circle has equation l O x x 2 +y 2 14x 8y +c =0. 2 Determine the value of c. (c)the line PQ is a diameter of the circle. Determine the coordinates ofq. 2 y y =x 3 3x 2 +5x +4 P C 16.Acirclehasequationx 2 +y 2 ax +2by +c =0,wherea,bandcareconstants. (a)giventhatthecentreofthecircleis ( 1,4),determinethevaluesofaandb. 2 (b)giventhatthepoint (5,4)liesonthecircle,determinethevalueofc. 2 17.AcirclehasP(0,1)andQ( 10,9)asendpointsofitsdiameter. Find the equation of this circle. 3 18.Findtheequationofthecirclecentredat (1, 3)andpassingthrough (7,5). Determine whether or not the point (8, 4) lies within this circle. 3 19. 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 hsn.uk.net Page 4 Questions marked c SQA
20. ThepointP(2,3)liesonthecircle (x +1) 2 + (y 1) 2 =13.Findtheequationof thetangentatp. 4 21.Acirclehasequationx 2 +y 2 8x +6y +15 =0. FindtheequationofthetangenttothecircleatthepointA(1, 2). 4 22. (a)find the equation of the line which passes through ( 1, 4) and is perpendiculartothelinewithequationx +3y =2. 3 (b)showthatthelinedoesnotintersectthecirclewithequation (x 2) 2 + (y +3) 2 =6. 3 23.Forwhichvaluesofkisthelinewithequationy= 3 4 x +katangenttothecircle withequationx 2 +y 2 =16? 5 24. The points A and B have coordinates (6, 8) and (18, 8) respectively. (a)findtheequationofl,theperpendicularbisectorofab. 4 (b)showthatlisatangenttothecirclewithequationx 2 +y 2 2x +4y 20 =0 and find the coordinates of the point of contact. 6 25.Forwhichvaluesofkdoestheequationx 2 +y 2 +10x 14y +k =0representa circle? 3 26.Findtherangeofvaluesofkforwhichtheequationx 2 +y 2 kx (k+1)y+ 5 4 =0 represents a circle. 6 27. Forwhatrangeofvaluesofkdoestheequationx 2 +y 2 +4kx 2ky k 2 =0 represent a circle? 5 28.Thecirclewithcentre (8,a)passesthroughthepoints (3,3)and (5,1). Findthevalueofaandhencefindtheequationofthecircle. 5 hsn.uk.net Page 5 Questions marked c SQA
29. (a)findtheradiusandequationofthecirclecentredattheoriginandpassing through the point (3, 4). 2 (b) Determine whether the point (8, 5) lies inside, outside or on this circle. 2 30.Findtherangeofvaluesofkforwhich2x 2 +2y 2 6x +4y +k =0representsa circle. 4 [ENDOFPAPER1SECTIONB] hsn.uk.net Page 6 Questions marked c SQA
Paper 2 1. Find the equation of the tangent at the point (3, 4) on the circle x 2 +y 2 +2x 4y 15 =0. 4 2. Triangle ABC has vertices A(2, 2), B(12, 2) and C(8, 6). (a)writedowntheequationofl 1, the perpendicular bisector of AB. 1 A(2, 2) B(12, 2) (b)find the equation of l 2, the perpendicular bisector of AC. O x 4 (c) Find the point of intersection of linesl 1 andl 2. 1 (d)hencefindtheequationofthe circle passing through A, B and C. 2 y C(8, 6) 3. (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 hsn.uk.net Page 7 Questions marked c SQA
4. 5. FindtheequationofthecirclewhichhasP( 2, 1)andQ(4,5)astheendpoints of a diameter. 3 6. Theliney = 1isatangenttoacirclewhichpassesthrough (0,0)and (6,0). Find the equation of this circle. 6 7. hsn.uk.net Page 8 Questions marked c SQA
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10. 11. 12.Thecirclewithequationx 2 +y 2 10x +9 =0isshownbelow. y x 2 +y 2 10x +9 =0 O A x ThepointAliesonthecircumferenceofthecircleandthex-axis. FindtheequationofthetangenttothecircleatA. 5 hsn.uk.net Page 10 Questions marked c SQA
13. Find the equation of the tangent at the point (3, 1) on the circle x 2 +y 2 4x +6y 4 =0. 5 14. 15.Agardenisbeingdesignedsothattherearethree areas for planting around a triangular decked area. A diagram of the garden, relative to a set of coordinate axes is shown to the right. The points P, Q and R lie on the circle with equation x 2 +y 2 = 4. The triangle PQR is equilateral, and the line PR is parallel to the x-axis. (a)findtheequationofthelinethroughpand Q. 4 (b) (i)findthecoordinatesofthepointp. (ii) Hence calculate the shaded area. 7 P y O Q x 2 +y 2 =4 R x 16.Thelinewithequationy +x =5meetsthecirclewithequation x 2 +y 2 8x +2y 3 =0atthepointsPandQ. (a)findthecoordinatesofpandq. 4 (b)findtheequationofthecirclewhichhaspqasitsdiameter. 3 hsn.uk.net Page 11 Questions marked c SQA
17. 18.Thecirclewithequation (x 8) 2 + (y 4) 2 =16 isshowninthediagram. ThelineOAisanon-horizontaltangenttothe circle. (a)giventhatoahasgradientm,writedown the equation of OA. 1 (b) Hence or otherwise determine the value O ofm. x 7 y A hsn.uk.net Page 12 Questions marked c SQA
19. (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 20.Findthevaluesofkforwhichthelinex +y =kisatangenttothe circlex 2 +y 2 4x +2 =0. 5 21. Findthepossiblevaluesofkforwhichthelinex y =kisatangenttothecircle x 2 +y 2 =18. 5 hsn.uk.net Page 13 Questions marked c SQA
22.The larger circle shown in the diagram has equation y B x 2 +y 2 10x 4y +12 =0. Thesmallercirclehascentre (10, 13 4 )andradius 17 4. Thelinethrough AandBpassesthrough the centre of both circles. Point A has coordinates (1, 1). (a) Show that the circles touch externally. 4 (b) (i)findtheequationofthetangenttothelargercircleata. (ii)hencestatethegradientofthetangenttothesmallercircleatb. 4 O A x 23. (a)showthatthepointp(5,10)liesoncirclec 1 withequation (x +1) 2 + (y 2) 2 =100. 1 (b)pqisadiameterofthiscircleas y showninthediagram. Findthe equationofthetangentatq. P(5, 10) 5 O x Q (c)twocircles,c 2 andc 3,touchcircleC 1 atq. TheradiusofeachofthesecirclesistwicetheradiusofcircleC 1. FindtheequationsofcirclesC 2 andc 3. 4 24. hsn.uk.net Page 14 Questions marked c SQA
25. Explainwhytheequationx 2 +y 2 +2x +3y +5 =0doesnotrepresentacircle. 2 26. Forwhatrangeofvaluesofcdoestheequationx 2 +y 2 6x +4y+c =0represent acircle? 3 27. hsn.uk.net Page 15 Questions marked c SQA
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34.Showthattheequationx 2 +y 2 +kx 4y 2 =0representsacircleforallvalues ofk. 3 35. 36.CircleC 1 hasequation (x +1) 2 + (y 1) 2 =121. AcircleC 2 withequationx 2 +y 2 4x +6y +p =0isdrawninsideC 1. The circles have no points of contact. Whatistherangeofvaluesofp? 9 [ENDOFPAPER2] hsn.uk.net Page 19 Questions marked c SQA