equations","content":"Similarly, comparing the given $y$-equation with the standard parametric equation allows us to find the $y$-coordinate of the centre $k$ and confirm the radius $r$:\n\n$$y = k + r \\sin \\theta$$\n\n$$y = -6 + 3 \\sin \\theta$$\n\nBy direct comparison, we find:\n- $k = -6$\n- $r = 3$"},{"type":"text","order":3,"title":"State final answers","content":"From our comparison, we have found:\n- $h = 1$\n- $k = -6$\n- $r = 3$\n\nTherefore:\n- The **centre** of the circle is **$(1, -6)$**.\n- The **radius** of the circle is **$3$**."}]}],"generatedAt":"2026-06-09T10:22:38.040Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"123a28d2-e83b-41f2-8140-bb62f2823ccd","revisionId":"99409257-2657-4fde-a2b1-da6b410eaeb0","questionNumber":"3","questionType":"LA","specification":"A circle is defined by the parametric equations $x = 5 + 2 \\cos \\theta$, $y = 3 + 2 \\sin \\theta$","difficulty":1,"difficultyBand":"easy","totalMarks":4,"sourceTitle":"Question Set 8: Parametric Equations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"aa54522c-6043-40c3-aab7-bcddfdd7304d","label":"","content":"Find the centre and the radius of the circle.","markscheme":"- Rearrange for trig: $\\cos \\theta = \\frac{x-5}{2}$ and $\\sin \\theta = \\frac{y-3}{2}$ M1\n- Use $\\cos^2 \\theta + \\sin^2 \\theta = 1$: $\\frac{(x-5)^2}{4} + \\frac{(y-3)^2}{4} = 1$ M1\n- Cartesian form: $(x-5)^2 + (y-3)^2 = 4$ A1\n- Centre $(5, 3)$, Radius $2$ A1","marks":4,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Rearranges equations for $\\cos \\theta$ and $\\sin \\theta$.","code":"M"},{"id":"p2","marks":1,"description":"Uses the identity $\\cos^2 \\theta + \\sin^2 \\theta = 1$ to eliminate the parameter.","code":"M"},{"id":"p3","marks":1,"description":"Correctly identifies centre $(5, 3)$ and radius $2$.","code":"A"}],"finalAnswers":["Centre (5, 3), Radius 2"],"generatedAt":"2026-05-06T03:46:57.180Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"cartesian_conversion","label":"Cartesian Equation Method","steps":[{"type":"text","order":0,"title":"Isolate trigonometric terms","content":"To convert the parametric equations into a single Cartesian equation, we first rearrange each equation to make $\\cos\\theta$ and $\\sin\\theta$ the subjects.\n\nFor the $x$-equation:\n$$x = 5 + 2\\cos\\theta$$\n$$x - 5 = 2\\cos\\theta$$\n$$\\cos\\theta = \\frac{x - 5}{2}$$\n\nFor the $y$-equation:\n$$y = 3 + 2\\sin\\theta$$\n$$y - 3 = 2\\sin\\theta$$\n$$\\sin\\theta = \\frac{y - 3}{2}$$\n\nThis step earns the first mark M1."},{"type":"text","order":1,"title":"Use Pythagorean identity","content":"Next, we use the fundamental trigonometric identity relating sine and cosine:\n\n$$\\sin^2\\theta + \\cos^2\\theta = 1$$\n\nSubstitute the expressions we found in the previous step into this identity:\n\n$$\\left(\\frac{y - 3}{2}\\right)^2 + \\left(\\frac{x - 5}{2}\\right)^2 = 1$$\n\nThis substitution earns the second mark M1."},{"type":"text","order":2,"title":"Form Cartesian equation","content":"Now, let's simplify the equation to match the standard form of a circle. Square the numerators and denominators:\n\n$$\\frac{(x - 5)^2}{4} + \\frac{(y - 3)^2}{4} = 1$$\n\nMultiply both sides of the equation by $4$ to clear the denominators:\n\n$$(x - 5)^2 + (y - 3)^2 = 4$$\n\nThis standard Cartesian form earns the third mark A1."},{"type":"text","order":3,"title":"Identify centre and radius","content":"The standard Cartesian equation of a circle is given by:\n\n$$(x - h)^2 + (y - k)^2 = r^2$$\n\nwhere $(h, k)$ is the centre and $r$ is the radius.\n\nComparing our equation $(x - 5)^2 + (y - 3)^2 = 4$ to the standard form:\n- The coordinates of the centre are $(h, k) = (5, 3)$.\n- The radius is $r = \\sqrt{4} = 2$.\n\nStating both values correctly earns the final mark A1."}]},{"id":"direct_comparison","labe

8.2 Using Trigonometric Identities

Question 2

8.2 Using Trigonometric Identities Questions

8.2 Using Trigonometric Identities

Question 2

8.2 Using Trigonometric Identities Questions