| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Tangent equation at a known point on circle |
| Difficulty | Moderate -0.3 This is a straightforward multi-part circle question requiring standard techniques: finding radius using distance formula, verifying a point satisfies the equation, and finding a tangent using perpendicular gradient. All parts are routine C1 procedures with no problem-solving insight needed, making it slightly easier than average but not trivial due to the multi-step nature. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| (i) radius \(= \sqrt{25 + 1} = \sqrt{26}\); \(\therefore (x+3)^2 + (y-2)^2 = (\sqrt{26})^2\); \((x+3)^2 + (y-2)^2 = 26\) | M1, A1; M1; A1 | |
| (ii) \((-4, 7)\): LHS \(= (-4+3)^2 + (7-2)^2 = 1 + 25 = 26\) \(\therefore\) lies on circle | B1 | |
| (iii) grad of radius \(= \frac{7-2}{-4-(-3)} = -5\); \(\therefore\) grad of tangent \(= -\frac{1}{5}\); \(\therefore y - 7 = \frac{1}{5}(x+4)\); \(5y - 35 = x + 4\); \(x - 5y + 39 = 0\) | M1; M1, A1; M1; A1 | (10) |
**(i)** radius $= \sqrt{25 + 1} = \sqrt{26}$; $\therefore (x+3)^2 + (y-2)^2 = (\sqrt{26})^2$; $(x+3)^2 + (y-2)^2 = 26$ | M1, A1; M1; A1 |
**(ii)** $(-4, 7)$: LHS $= (-4+3)^2 + (7-2)^2 = 1 + 25 = 26$ $\therefore$ lies on circle | B1 |
**(iii)** grad of radius $= \frac{7-2}{-4-(-3)} = -5$; $\therefore$ grad of tangent $= -\frac{1}{5}$; $\therefore y - 7 = \frac{1}{5}(x+4)$; $5y - 35 = x + 4$; $x - 5y + 39 = 0$ | M1; M1, A1; M1; A1 | (10)9. The circle $C$ has centre $( - 3,2 )$ and passes through the point $( 2,1 )$.\\
(i) Find an equation for $C$.\\
(ii) Show that the point with coordinates $( - 4,7 )$ lies on $C$.\\
(iii) Find an equation for the tangent to $C$ at the point ( - 4 , 7). Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.\\
\hfill \mbox{\textit{OCR C1 Q9 [10]}}