| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Circles |
| Type | Circle from diameter endpoints |
| Difficulty | Moderate -0.8 This is a straightforward vectors and circle question requiring only recognition of standard geometric interpretations (distance, midpoint, circle equation) and routine algebraic manipulation. All parts are direct applications of definitions with no problem-solving or novel insight required, making it easier than average. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement |
$A$ and $B$ are fixed points in the $x$-$y$ plane. The position vectors of $A$ and $B$ are $\mathbf{a}$ and $\mathbf{b}$ respectively.
State, with reference to points $A$ and $B$, the geometrical significance of
\begin{enumerate}[label=(\alph*)]
\item the quantity $|\mathbf{a} - \mathbf{b}|$, [1]
\item the vector $\frac{1}{2}(\mathbf{a} + \mathbf{b})$. [1]
\end{enumerate}
The circle $P$ is the set of points with position vector $\mathbf{p}$ in the $x$-$y$ plane which satisfy
$$\left|\mathbf{p} - \frac{1}{2}(\mathbf{a} + \mathbf{b})\right| = \frac{1}{2}|\mathbf{a} - \mathbf{b}|.$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item State, in terms of $\mathbf{a}$ and $\mathbf{b}$,
\begin{enumerate}[label=(\roman*)]
\item the position vector of the centre of $P$, [1]
\item the radius of $P$. [1]
\end{enumerate}
\end{enumerate}
It is now given that $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}$ and $\mathbf{p} = \begin{pmatrix} x \\ y \end{pmatrix}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find a cartesian equation of $P$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2020 Q7 [8]}}