tex":"\\htmlData{anchor=sub2}{(x-10)^2+\\left(-\\frac14x-6\\right)^2=68}"},{"latex":"\\htmlData{anchor=quad2}{\\frac{17}{16}x^2-17x+68=0}"},{"latex":"\\htmlData{anchor=factor2}{\\frac{17}{16}(x-8)^2=0}"},{"latex":"\\htmlData{anchor=root2}{x=8\\quad\\text{(M1)}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"root2"}],"connections":[{"to":"sub2","from":"tan2","color":"sky","label":"substitute for y"},{"to":"quad2","from":"sub2","color":"amber","label":"expand and simplify"},{"to":"factor2","from":"quad2","color":"green","label":"factorise"},{"to":"root2","from":"factor2","color":"teal","label":"solve"}]},"order":3,"title":"Find the second contact","description":"Now use the tangent with gradient negative one quarter. Substitute into the same circle equation and factorise the repeated-root quadratic."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=xtwo}{\\color{@sky}{x=8}}"},{"latex":"\\htmlData{anchor=yminus}{\\color{@green}{y=-\\frac14(8)=-2}}"},{"latex":"\\htmlData{anchor=pointtwo}{\\color{@amber}{(8,-2)\\quad\\text{(A1)}}}"}],"highlights":[{"color":"amber","style":"text-color-with-background","anchor":"pointtwo"}],"connections":[{"to":"yminus","from":"xtwo","color":"sky","label":"use tangent equation"},{"to":"pointtwo","from":"yminus","color":"amber","label":"form point"}]},"order":4,"title":"Get the second point","description":"Put the repeated x-value into the second tangent equation, then combine the two coordinates."}]}],"generatedAt":"2026-07-21T04:27:24.460Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"adcfd298-9ebc-499f-bcb4-2e49bf761893","revisionId":"766cb928-ff43-4477-bde1-84ababe218df","questionNumber":"19","questionType":"LA","specification":"The lines with equations $y = \\frac{2}{3}x$ and $y = -\\frac{2}{3}x$ are tangents to a circle at $(3, 2)$ and $(3, -2)$ respectively.","difficulty":4,"difficultyBand":"hard","totalMarks":6,"sourceTitle":"Question Set 6: Circles","calculatorUse":null,"isCalculation":false,"parts":[{"id":"f7fec70b-1bae-44ad-b33b-a8c891f7a218","label":"0","content":"Find an equation for the circle.","markscheme":"- Gradient of radius at $(3, 2)$ is $-\\frac{3}{2}$. Equation: $y - 2 = -\\frac{3}{2}(x - 3) \\Rightarrow 3x + 2y = 13$ M1\n- Gradient of radius at $(3, -2)$ is $\\frac{3}{2}$. Equation: $y + 2 = \\frac{3}{2}(x - 3) \\Rightarrow 3x - 2y = 13$ M1\n- Solve simultaneously: $x = \\frac{13}{3}, y = 0$. Centre is $(\\frac{13}{3}, 0)$ M1\n- $r^2 = (3 - \\frac{13}{3})^2 + 2^2 = (-\\frac{4}{3})^2 + 4 = \\frac{16}{9} + \\frac{36}{9} = \\frac{52}{9}$ M1\n- Equation: $(x - \\frac{13}{3})^2 + y^2 = \\frac{52}{9}$ A2","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Finds equation of one radius line.","code":"M"},{"id":"p2","marks":1,"description":"Finds equation of the second radius line.","code":"M"},{"id":"p3","marks":1,"description":"Solves for the centre of the circle.","code":"M"},{"id":"p4","marks":1,"description":"Calculates $r^2$.","code":"M"},{"id":"p5","marks":2,"description":"Correct final equation $(x - \\frac{13}{3})^2 + y^2 = \\frac{52}{9}$.","code":"A"}],"finalAnswers":["(x - \\frac{13}{3})^2 + y^2 = \\frac{52}{9}"],"generatedAt":"2026-05-12T02:14:06.605Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"perpendicular_radii","label":"Intersect the perpendicular radii","steps":[{"type":"text","order":0,"title":"Use perpendicular gradients","content":"A radius meets a tangent at a right angle. Therefore, its gradient is the **negative reciprocal** of the tangent's gradient: change the sign and turn the fraction upside down.\n\nAt $(3,2)$, the tangent gradient is $\\frac{2}{3}$, so the radius gradient is $-\\frac{3}{2}$. At $(3,-2)$, the tangent gradient is $-\\frac{2}{3}$, so the radius gradient is $\\frac{3}{2}$.","highlights":[{"text":"tang

Circles

Question 3

Circles Questions

Circles

Question 3

Circles Questions