| Exam Board | OCR MEI |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Circle touching axes |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question requiring basic circle concepts. Part (a) uses the fact that if the x-axis is tangent, the radius equals the y-coordinate (4 units), giving equation (x-10)² + (y-4)² = 16. Part (b) requires substituting y=x and checking the discriminant or using perpendicular distance formula. Part (c) is trivial coordinate geometry. All parts are routine applications with no problem-solving insight needed, making this easier than average. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03f Circle properties: angles, chords, tangents1.10e Position vectors: and displacement |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Radius \(= 4\) | B1 | soi by 16 |
| \((x-10)^2 + (y-4)^2 = 16\) | B1 | o.e. |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Where \(y = x\) meets the circle: \((x-10)^2 + (x-4)^2 = 16\) | M1 | For sub'n of \(y = x\) into their circle equation |
| \(2x^2 - 28x + 116 = 16\) or equivalent | M1 | For expanding and collecting like terms |
| \(x^2 - 14x + 50 = 0\) or \(2x^2 - 28x + 100 = 0\) | M1 | Rearranging to 3 term quadratic \(= 0\) |
| \(b^2 - 4ac = -4\) or \(-16\) so no meeting points i.e. not a tangent | E1 | Or \((x-7)^2 + 1 = 0\) no meeting points hence not a tangent; 'No real roots' not enough for E1; Allow E1 for correct reasoning following omission of '\(=0\)' |
| Alternative method: Angle between \(y=x\) and \(x\)-axis \(= 45°\); Let \(\theta\) be angle between \(x\)-axis and line joining \((0,0)\) and \((10,4)\), \(\tan\theta = 0.4\); \(\theta = 21.80°\) (2 d.p.); If \(y=x\) was a tangent \(\theta\) would be \(22.5°\) hence not a tangent | M1, M1, M1, E1 | |
| Alternative method 2: gradient of normal is \(-1\); Line joining point of contact \((k,k)\) to centre \((10,4)\) must have gradient \(-1\); \(k=7\); \((7,7)\) does not lie on circle | ||
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(5\mathbf{i} + 2\mathbf{j}\) or \(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\) | B1 | Must be correct vector notation; not \((5, 2)\) |
| [1] |
## Question 6:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Radius $= 4$ | B1 | soi by 16 |
| $(x-10)^2 + (y-4)^2 = 16$ | B1 | o.e. |
| **[2]** | | |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Where $y = x$ meets the circle: $(x-10)^2 + (x-4)^2 = 16$ | M1 | For sub'n of $y = x$ into their circle equation |
| $2x^2 - 28x + 116 = 16$ or equivalent | M1 | For expanding and collecting like terms |
| $x^2 - 14x + 50 = 0$ or $2x^2 - 28x + 100 = 0$ | M1 | Rearranging to 3 term quadratic $= 0$ |
| $b^2 - 4ac = -4$ or $-16$ so no meeting points i.e. not a tangent | E1 | Or $(x-7)^2 + 1 = 0$ no meeting points hence not a tangent; 'No real roots' not enough for E1; Allow E1 for correct reasoning following omission of '$=0$' |
| **Alternative method:** Angle between $y=x$ and $x$-axis $= 45°$; Let $\theta$ be angle between $x$-axis and line joining $(0,0)$ and $(10,4)$, $\tan\theta = 0.4$; $\theta = 21.80°$ (2 d.p.); If $y=x$ was a tangent $\theta$ would be $22.5°$ hence not a tangent | M1, M1, M1, E1 | |
| **Alternative method 2:** gradient of normal is $-1$; Line joining point of contact $(k,k)$ to centre $(10,4)$ must have gradient $-1$; $k=7$; $(7,7)$ does not lie on circle | | |
| **[4]** | | |
### Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $5\mathbf{i} + 2\mathbf{j}$ or $\begin{pmatrix} 5 \\ 2 \end{pmatrix}$ | B1 | Must be correct vector notation; not $(5, 2)$ |
| **[1]** | | |
---6 A circle has centre $C ( 10,4 )$. The $x$-axis is a tangent to the circle, as shown in Fig. 6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{99485c27-9ff8-4bdb-a7e6-49dfcaedc579-5_605_828_979_255}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Find the equation of the circle.
\item Show that the line $y = x$ is not a tangent to the circle.
\item Write down the position vector of the midpoint of OC.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 3 2019 Q6 [7]}}