| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Distance from centre to line |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring completing the square to find circle centre/radius (standard C1 technique), then finding the perpendicular distance from centre to line using basic coordinate geometry. All steps are routine textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((x-1)^2 + (y-9)^2 = 9\) | M1 A2 | |
| Centre \((1,9)\), radius \(= 3\) | ||
| (ii) Gradient of line \(= 2\) | M1 | |
| Perpendicular gradient \(= -\frac{1}{2}\) | M1 | |
| Equation of line through centre of circle, perpendicular to straight line: \(y - 9 = -\frac{1}{2}(x-1)\) | M1 | |
| \(y = \frac{19}{2} - \frac{1}{2}x\) | A1 | |
| Closest point where lines intersect: \(2x - 3 = \frac{19}{2} - \frac{1}{2}x\) | M1 | |
| \(x = 5\) | A1 | |
| \(\therefore (5,7)\) | A1 | (9) |
**(i)** $(x-1)^2 + (y-9)^2 = 9$ | M1 A2 |
Centre $(1,9)$, radius $= 3$ | |
**(ii)** Gradient of line $= 2$ | M1 |
Perpendicular gradient $= -\frac{1}{2}$ | M1 |
Equation of line through centre of circle, perpendicular to straight line: $y - 9 = -\frac{1}{2}(x-1)$ | M1 |
$y = \frac{19}{2} - \frac{1}{2}x$ | A1 |
Closest point where lines intersect: $2x - 3 = \frac{19}{2} - \frac{1}{2}x$ | M1 |
$x = 5$ | A1 |
$\therefore (5,7)$ | A1 | **(9)**
---8.
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\includegraphics[max width=\textwidth, alt={}]{4fa65854-801c-4a93-866e-796c000a649f-2_675_689_251_495}
\end{center}
The diagram shows the circle with equation $x ^ { 2 } + y ^ { 2 } - 2 x - 18 y + 73 = 0$ and the straight line with equation $y = 2 x - 3$.\\
(i) Find the coordinates of the centre and the radius of the circle.\\
(ii) Find the coordinates of the point on the line which is closest to the circle.\\
\hfill \mbox{\textit{OCR C1 Q8 [9]}}