}],"highlights":[{"color":"purple","style":"text-color-with-background","anchor":"r2"}],"connections":[{"to":"diff","from":"setup","color":"purple","label":"simplify differences"},{"to":"sq","from":"diff","color":"purple","label":"square"},{"to":"r2","from":"sq","color":"purple","label":"add"}]},"order":5,"title":"Calculate the radius","description":"Use the squared distance from the centre to a point of contact. Correctly finding the squared radius earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=circle_form}{(x-a)^2+(y-b)^2=r^2}$"},{"latex":"$$\\htmlData{anchor=circle_substitution}{\\left(x-\\frac{13}{3}\\right)^2+(y-0)^2=\\frac{52}{9}}$"},{"latex":"$$\\htmlData{anchor=final_circle}{\\left(x-\\frac{13}{3}\\right)^2+y^2=\\frac{52}{9}}$"}],"highlights":[{"color":"teal","style":"text-color-with-background","anchor":"final_circle"}],"connections":[{"to":"circle_substitution","from":"circle_form","color":"teal","label":"substitute centre and radius"},{"to":"final_circle","from":"circle_substitution","color":"teal","label":"simplify"}]},"order":6,"title":"Write the equation","description":"Use the standard circle equation, then substitute the centre and squared radius. Both correct values earn the final A2."}]}],"generatedAt":"2026-07-20T16:58:51.544Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"bdb981cd-5a19-4f3c-8764-ace97b6ffd74","revisionId":"3cf8071d-fa90-44d1-8f15-1e067788f34f","questionNumber":"27","questionType":"LA","specification":"A circle $C$ with centre $(2, 1)$ passes through the point $(-1, 3)$. A line $L$ passes through the points $(1, -4)$ and $(10, 2)$.","difficulty":4,"difficultyBand":"hard","totalMarks":7,"sourceTitle":"Question Set 6: Circles","calculatorUse":null,"isCalculation":true,"parts":[{"id":"e1bdadd1-5d65-4e7e-9674-ca0ee3204991","label":"0","content":"Show that $L$ is a tangent to $C$.","markscheme":"- Circle equation: $(x - 2)^2 + (y - 1)^2 = (-1 - 2)^2 + (3 - 1)^2 = 13$ M1\n- Line gradient: $m = \\frac{2 - (-4)}{10 - 1} = \\frac{6}{9} = \\frac{2}{3}$ M1\n- Line equation: $y - 2 = \\frac{2}{3}(x - 10) \\Rightarrow y = \\frac{2}{3}x - \\frac{14}{3}$ M1\n- Substitute line into circle: $(x - 2)^2 + (\\frac{2}{3}x - \\frac{14}{3} - 1)^2 = 13$ M1\n- Expand and simplify to $13x^2 - 104x + 208 = 0 \\Rightarrow x^2 - 8x + 16 = 0$ M1\n- $(x - 4)^2 = 0$. Since there is only one solution ($x = 4$), the line is a tangent. A2","marks":7,"order":0,"markingRubric":{"totalMarks":7,"points":[{"id":"p1","marks":1,"description":"Finds the equation of the circle.","code":"M"},{"id":"p2","marks":1,"description":"Finds the equation of the line $L$.","code":"M"},{"id":"p3","marks":1,"description":"Substitutes the line equation into the circle equation.","code":"M"},{"id":"p4","marks":1,"description":"Simplifies to a quadratic equation in $x$.","code":"M"},{"id":"p5","marks":1,"description":"Shows the quadratic has only one root (e.g., by factoring or discriminant).","code":"M"},{"id":"p6","marks":2,"description":"Concludes that $L$ is a tangent because there is exactly one intersection point.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-12T02:14:06.605Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"repeated_intersection","label":"Find the repeated intersection","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$r^2=\\htmlData{anchor=radiuscalc}{(-1-2)^2+(3-1)^2}$"},{"latex":"$$\\htmlData{anchor=radiusvalue}{r^2}=13$"},{"latex":"$$(x-2)^2+(y-1)^2=\\htmlData{anchor=circlerhs}{13}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"radiuscalc"}],"connections":[{"to":"radiusvalue","from":"radiuscalc","color":"sky","label":"evaluate"},{"to":"circlerhs","from":"radiusvalue","color":"sky","label":"use in equation"}]},"order":0,"title":"Find the circle","description":"Use the given point to find the squared rad

1.6 C: Coordinate geometry in the (x, y) plane

Question 4

1.6 C: Coordinate geometry in the (x, y) plane Questions

1.6 C: Coordinate geometry in the (x, y) plane

Question 4

1.6 C: Coordinate geometry in the (x, y) plane Questions