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M1\n- Centre: $(1,-3)$. A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Completes the square to rewrite the equation as $(x-1)^2+(y+3)^2=36$.","code":"M"},{"id":"p2","marks":1,"description":"States that the centre is $(1,-3)$.","code":"A"}],"finalAnswers":["$$(1,-3)$"],"generatedAt":"2026-08-31T10:09:05.586Z","modelVersion":"cheap:gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]},{"id":"d922a594-15a7-4946-adb8-e4de394f109c","label":"ii","content":"Find the radius of $C$.","markscheme":"- From $(x-1)^2+(y+3)^2=36$, the radius is $6$. B1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"States that the radius is $6$.","code":"B"}],"finalAnswers":["$$6$"],"generatedAt":"2026-08-31T10:09:05.586Z","modelVersion":"cheap:gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"c0c880bd-08f7-4181-a9f3-2e9487df1b79","revisionId":"7f99253f-2605-4203-9323-cdf1dd216140","questionNumber":"2","questionType":"LA","specification":"The circle $C$ has centre $(2,5)$ and passes through the point $(4,9)$.","difficulty":1,"difficultyBand":null,"totalMarks":3,"sourceTitle":"6.2 Equation of a Circle","calculatorUse":null,"isCalculation":false,"parts":[{"id":"4e1941ed-d294-401a-ac2a-de6a33176f76","label":"","content":"Find an equation for $C$.","markscheme":"- Use $(x-2)^2+(y-5)^2=r^2$. M1\n- Substitute $(4,9)$ to obtain $r^2=(4-2)^2+(9-5)^2=20$. A1\n- Equation: $(x-2)^2+(y-5)^2=20$. A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Uses the circle form $(x-2)^2+(y-5)^2=r^2$.","code":"M"},{"id":"p2","marks":1,"description":"Substitutes $(4,9)$ and obtains $r^2=20$.","code":"A"},{"id":"p3","marks":1,"description":"Gives the equation $(x-2)^2+(y-5)^2=20$.","code":"A","acceptable":["$$x^2+y^2-4x-10y+9=0$"]}],"finalAnswers":["$$(x-2)^2+(y-5)^2=20$"],"generatedAt":"2026-08-31T10:09:05.586Z","modelVersion":"cheap:gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"f7cf600e-b1a7-4dc6-bb3a-1b060decc50d","revisionId":"c15e67b6-9e1e-4508-ae1f-eb690b4d56e9","questionNumber":"3","questionType":"LA","specification":"The circle $C$ has centre $(-2,3)$ and passes through the point $(1,8)$.","difficulty":2,"difficultyBand":null,"totalMarks":10,"sourceTitle":"6.2 Equation of a Circle","calculatorUse":null,"isCalculation":true,"parts":[{"id":"150da1ec-564f-4765-9752-d5c68c8b8871","label":"a","content":"Find an equation for $C$.","markscheme":"- Use $(x+2)^2+(y-3)^2=r^2$. M1\n- Substitute $(1,8)$ into the equation. M1\n- Obtain $r^2=(1+2)^2+(8-3)^2=34$. A1\n- Equation: $(x+2)^2+(y-3)^2=34$. A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Uses the circle form $(x+2)^2+(y-3)^2=r^2$.","code":"M"},{"id":"p2","marks":1,"description":"Substitutes the point $(1,8)$ into the circle equation.","code":"M"},{"id":"p3","marks":1,"description":"Calculates $r^2=34$.","code":"A"},{"id":"p4","marks":1,"description":"Gives the equation $(x+2)^2+(y-3)^2=34$.","code":"A","acceptable":["$$x^2+y^2+4x-6y-21=0$"]}],"finalAnswers":["$$(x+2)^2+(y-3)^2=34$"],"generatedAt":"2026-08-31T10:09:05.586Z","modelVersion":"cheap:gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]},{"id":"8a2e00eb-0238-4926-bb25-d4eecc1e6160","label":"b","content":"Show that the point $(3,6)$ lies on $C$.","markscheme":"- Substitute $(3,6)$ to show that $(3+2)^2+(6-3)^2=25+9=34$. B1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Substitutes $(3,6)$ and shows that the left-hand side equals $34$.","code":"B","acceptable":["$$(3+2)^2+(6-3)^2=34$"]}],"finalAnswers":[],"generatedAt":"2026-08-31T10:09:05.586Z","modelVersion":"cheap:gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]},{"id":"da7d6be6-09e2-41b3-b637-bf2ccf1f2d7d","label":"c","content":"Find the equation of the tangent to $C$ at $(3,6)$.\n\nGive your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.","markscheme":"- Calculate the gradient of the radius as $\\dfrac{6-3}{3-(-2)}=\\dfrac35$. M1\n- Use perpendicular gradients to obtain the tangent gradient $-\\dfrac53$. A1\n- Use the point $(3,6)$ in a line equation with gradient $-\\dfrac53$. M1\n- Obtain $y=-\\dfrac53x+11$. A1\n- Equation: $5x+3y-33=0$. A1","marks":5,"order":2,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Calculates the gradient of the radius as $\\frac35$.","code":"M"},{"id":"p2","marks":1,"description":"Finds the perpendicular tangent gradient $-\\frac53$.","code":"A"},{"id":"p3","marks":1,"description":"Uses the point $(3,6)$ in a line equation with gradient $-\\frac53$.","code":"M","acceptable":["$$y-6=-\\frac53(x-3)$"]},{"id":"p4","marks":1,"description":"Obtains the equivalent equation $y=-\\frac53x+11$.","code":"A"},{"id":"p5","marks":1,"description":"Gives the tangent as $5x+3y-33=0$.","code":"A"}],"finalAnswers":["$$5x+3y-33=0$"],"generatedAt":"2026-08-31T10:09:05.586Z","modelVersion":"cheap:gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"as_level_maths","summaries":[{"id":"d19e70ad-36e1-4a7f-9ba8-5e36254d372a","difficultyBand":null,"calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c0c880bd-08f7-4181-a9f3-2e9487df1b79","difficultyBand":null,"calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"f7cf600e-b1a7-4dc6-bb3a-1b060decc50d","difficultyBand":null,"calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"c3d39fec-b4ab-4584-a3b7-2580dc92e245","difficultyBand":null,"calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"4f2a67b5-bae2-445a-8142-19b4a2d173d2","difficultyBand":null,"calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"0295e06f-cc77-42a7-b462-43e3c46b157b","difficultyBand":null,"calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"242b882e-5979-4977-82b6-ce976ca41c2e","difficultyBand":null,"calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b62d1926-22ad-4dbe-816b-229da34a82b0","difficultyBand":null,"calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"661b0e02-c987-4504-b533-920014d77163","difficultyBand":null,"calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":false},{"id":"1eb8b815-3b8f-46c2-ad37-6d976074c692","difficultyBand":null,"calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false}],"title":"6.2 Equation of a Circle","topicId":"db0e283e-5dda-4962-ac56-45b0a2a9a15d","topicSidebar":"$L3b","topicSlug":"as-pure-mathematics-6-2-equation-of-a-circle","topicTitle":"6.2 Equation of a Circle"}]}],"$L3c","$L3d"]}]