21c96443423","questionNumber":"21","questionType":"LA","specification":"The diagram shows line with equation $y + 3x = 2$ which is a tangent to a circle with centre $(11, -1)$.\n\n![](https://assets.mathsgenie.co.uk/booklet/9d3d35ee5a86e541bf519c5a0858f122f2f9fbc5ca437c8330d8dad8fc80ac92.jpg)","difficulty":3,"difficultyBand":"medium","totalMarks":5,"sourceTitle":"Question Set 6: Circles","calculatorUse":null,"isCalculation":false,"parts":[{"id":"66d03d9e-5ee7-406f-906d-29ba1d16077a","label":"0","content":"Find an equation for the circle.","markscheme":"- Gradient of tangent is $-3$, so gradient of radius is $\\frac{1}{3}$ M1\n- Equation of radius line: $y + 1 = \\frac{1}{3}(x - 11) \\Rightarrow x - 3y = 14$ M1\n- Intersection of $x - 3y = 14$ and $3x + y = 2$: $x = 2, y = -4$ M1\n- $r^2 = (11 - 2)^2 + (-1 - (-4))^2 = 9^2 + 3^2 = 90$ M1\n- Equation: $(x - 11)^2 + (y + 1)^2 = 90$ A1\n![](https://assets.mathsgenie.co.uk/booklet/a3ddcadfb9b6b80e12e09b02c0a03f89331b12dc9c0e88fd38640038ab6465b1.jpg)","marks":5,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Finds the gradient and equation of the radius line.","code":"M"},{"id":"p2","marks":1,"description":"Finds the point of tangency by solving simultaneous equations.","code":"M"},{"id":"p3","marks":1,"description":"Calculates $r^2$ correctly.","code":"M"},{"id":"p4","marks":1,"description":"Correct final equation $(x - 11)^2 + (y + 1)^2 = 90$.","code":"A"}],"finalAnswers":["(x - 11)^2 + (y + 1)^2 = 90"],"generatedAt":"2026-05-12T02:14:06.605Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"perpendicular_radius","label":"Use the perpendicular radius","steps":[{"type":"text","order":0,"title":"Use the tangent property","content":"A radius drawn to the point of tangency is perpendicular to the tangent. Perpendicular gradients multiply to $-1$, so the tangent gradient $-3$ gives the radius gradient $\\frac{1}{3}$. This earns the first method mark.","highlights":[{"text":"line with equation $y + 3x = 2$ which is a tangent to a circle with centre $(11, -1)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=first_eq}{\\color{@sky}{y-(-1)=\\frac{1}{3}(x-11)}}$"},{"latex":"$$\\htmlData{anchor=second_eq}{\\color{@amber}{3(y+1)=x-11}}$"},{"latex":"$$\\htmlData{anchor=third_eq}{\\color{@green}{x-3y=14}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"first_eq"},{"color":"amber","style":"text-color","anchor":"second_eq"},{"color":"green","style":"text-color-with-background","anchor":"third_eq"}],"connections":[{"to":"second_eq","from":"first_eq","color":"green","label":"multiply by 3"},{"to":"third_eq","from":"second_eq","color":"green","label":"rearrange"}]},"order":1,"title":"Form the radius line","description":"The radius is perpendicular to the tangent, so its gradient is the negative reciprocal of negative three, namely one third. The radius has gradient 1/3 and passes through the centre (11, -1). Use point-gradient form, then rearrange. Reaching x - 3y = 14 earns the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=radius_text}{x-3y=14}\\htmlData{anchor=radius_right}{\\phantom{\\qquad}}$"},{"latex":"$$\\htmlData{anchor=rearranged_left}{\\phantom{\\qquad}}\\htmlData{anchor=rearranged_text}{x=14+3y}\\htmlData{anchor=rearranged_right}{\\phantom{\\qquad}}$"},{"latex":"$$\\htmlData{anchor=substituted_left}{\\phantom{\\qquad}}\\htmlData{anchor=substituted_text}{3(14+3y)+y=2}$"},{"latex":"$$42+10y=2$"},{"latex":"$$\\htmlData{anchor=y_text}{y=-4}\\htmlData{anchor=y_right}{\\phantom{\\qquad}}$"},{"latex":"$$\\htmlData{anchor=x_text}{x=14+3(-4)=2}\\htmlData{anchor=x_right}{\\phantom{\\qquad}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"radius_text"},{"color":"amber","style":"text-color","anchor":"rearranged_text"},{"color":"orange","style":"text-color","anchor":"substituted_text"},{"color":"green","style":"te

Circles

Question 16

Circles Questions

Circles

Question 16

Circles Questions