| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Circle from diameter endpoints |
| Difficulty | Moderate -0.3 This is a straightforward C2 circle question requiring standard techniques: finding circle equation from diameter endpoints, finding x-intercepts, and using the perpendicular radius property for tangents. All steps are routine applications of formulas with no novel problem-solving required, making it slightly easier than average. |
| 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 |
|---|---|---|
| (a) centre \(= (2, 3)\) | B1 | |
| radius \(= \sqrt{4 + 9} = \sqrt{13}\) | M1 | |
| \(\therefore (x-2)^2 + (y-3)^2 = (\sqrt{13})^2\) | M1 | |
| \((x-2)^2 + (y-3)^2 = 13\) | A1 | |
| (b) \(y = 0 \quad \therefore (x-2)^2 + 9 = 13\) | M1 | |
| \(x = 2 \pm \sqrt{4} = 0\) (at O) or \(4 \quad \therefore B(4,0)\) | A1 | |
| (c) grad of radius \(= \frac{0-3}{4-2} = -\frac{3}{2}\) | M1 | |
| \(\therefore\) grad of tangent \(= \frac{-1}{-\frac{3}{2}} = \frac{2}{3}\) | M1 A1 | |
| \(\therefore y - 0 = \frac{2}{3}(x - 4)\) | M1 | |
| \(3y = 2x - 8\) | ||
| \(2y - 3y = 8\) | A1 | (11) |
**(a)** centre $= (2, 3)$ | B1 |
radius $= \sqrt{4 + 9} = \sqrt{13}$ | M1 |
$\therefore (x-2)^2 + (y-3)^2 = (\sqrt{13})^2$ | M1 |
$(x-2)^2 + (y-3)^2 = 13$ | A1 |
**(b)** $y = 0 \quad \therefore (x-2)^2 + 9 = 13$ | M1 |
$x = 2 \pm \sqrt{4} = 0$ (at O) or $4 \quad \therefore B(4,0)$ | A1 |
**(c)** grad of radius $= \frac{0-3}{4-2} = -\frac{3}{2}$ | M1 |
$\therefore$ grad of tangent $= \frac{-1}{-\frac{3}{2}} = \frac{2}{3}$ | M1 A1 |
$\therefore y - 0 = \frac{2}{3}(x - 4)$ | M1 |
$3y = 2x - 8$ | |
$2y - 3y = 8$ | A1 | **(11)**\begin{enumerate}
\item The point $A$ has coordinates ( 4,6 ).
\end{enumerate}
Given that $O A$, where $O$ is the origin, is a diameter of circle $C$,\\
(a) find an equation for $C$.
Circle $C$ crosses the $x$-axis at $O$ and at the point $B$.\\
(b) Find the coordinates of $B$.\\
(c) Find an equation for the tangent to $C$ at $B$, giving your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers.\\
\hfill \mbox{\textit{Edexcel C2 Q8 [11]}}