| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Tangent equation involving finding the point of tangency |
| Difficulty | Standard +0.3 This is a standard C2 circle question requiring completion of the square, finding radius, x-intercepts, and using perpendicular gradient property for tangents. All techniques are routine for this level, though part (d) requires recognizing that AT is perpendicular to the tangent, making it slightly above average difficulty. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Centre \((5, 0)\) (or \(x=5\), \(y=0\)) | B1, B1 | (0,5) scores B1 B0; (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((x \pm a)^2 \pm b \pm 9 + (y \pm c)^2 = 0 \Rightarrow r^2 = \ldots\) or \(r = \ldots\), Radius \(= 4\) | M1, A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((1, 0)\), \((9, 0)\) | B1ft, B1ft | Allow just \(x=1\), \(x=9\); (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Gradient of \(AT = -\frac{2}{7}\) | B1 | |
| \(y = -\frac{2}{7}(x-5)\) | M1, A1ft | (3 marks) M1: line through centre, any gradient except 0 or \(\infty\); A1ft: follow through from centre but gradient must be \(-\frac{2}{7}\) |
## Question 8:
**(a)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Centre $(5, 0)$ (or $x=5$, $y=0$) | B1, B1 | (0,5) scores B1 B0; (2 marks) |
**(b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(x \pm a)^2 \pm b \pm 9 + (y \pm c)^2 = 0 \Rightarrow r^2 = \ldots$ or $r = \ldots$, Radius $= 4$ | M1, A1 | (2 marks) |
**(c)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1, 0)$, $(9, 0)$ | B1ft, B1ft | Allow just $x=1$, $x=9$; (2 marks) |
**(d)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Gradient of $AT = -\frac{2}{7}$ | B1 | |
| $y = -\frac{2}{7}(x-5)$ | M1, A1ft | (3 marks) M1: line through centre, any gradient except 0 or $\infty$; A1ft: follow through from centre but gradient must be $-\frac{2}{7}$ |
---8. The circle $C$, with centre at the point $A$, has equation $x ^ { 2 } + y ^ { 2 } - 10 x + 9 = 0$.
Find
\begin{enumerate}[label=(\alph*)]
\item the coordinates of $A$,
\item the radius of $C$,
\item the coordinates of the points at which $C$ crosses the $x$-axis.
Given that the line $l$ with gradient $\frac { 7 } { 2 }$ is a tangent to $C$, and that $l$ touches $C$ at the point $T$,
\item find an equation of the line which passes through $A$ and $T$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2005 Q8 [9]}}