| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Find parameter values for tangency using discriminant |
| Difficulty | Moderate -0.3 This is a comprehensive but routine C1 circle question covering standard techniques: completing the square to find centre/radius, using Pythagoras for chord distance, substituting to form a quadratic, and applying the discriminant condition for tangency. All parts follow textbook methods with no novel insight required, making it slightly easier than average despite being multi-part. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
7 A circle has equation $x ^ { 2 } + y ^ { 2 } - 4 x - 14 = 0$.
\begin{enumerate}[label=(\alph*)]
\item Find:
\begin{enumerate}[label=(\roman*)]
\item the coordinates of the centre of the circle;
\item the radius of the circle in the form $p \sqrt { 2 }$, where $p$ is an integer.
\end{enumerate}\item A chord of the circle has length 8. Find the perpendicular distance from the centre of the circle to this chord.
\item A line has equation $y = 2 k - x$, where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinate of any point of intersection of the line and the circle satisfies the equation
$$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$
\item Find the values of $k$ for which the equation
$$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$
has equal roots.
\item Describe the geometrical relationship between the line and the circle when $k$ takes either of the values found in part (c)(ii).
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2006 Q7 [17]}}