| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Tangent equation at a known point on circle |
| Difficulty | Moderate -0.8 This is a straightforward multi-part circle question testing standard techniques: completing the square to find centre/radius, distance calculations, and finding a tangent perpendicular to a radius. All parts are routine applications of well-practiced methods with clear signposting, making it easier than average for A-level. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x-4)^2 - 16 + (y-2)^2 - 4 = 9\) o. | M2 | M1 for one completing square or for \((x-4)^2\) or \((y-2)^2\) expanded correctly, or starting with \((x-4)^2 + (y-2)^2 = r^2\): M1 for correct expn of at least one bracket and M1 for \(9 + 20 = r^2\) o.e. |
| \(\text{rad} = \sqrt{29}\) | B1 | or using \(x^2 - 2gx + y^2 - 2fy + c = 0\): M1 for using centre is \((g, f)\) [must be quoted] and M1 for \(r^2 = g^2 + f^2 - c\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4^2 + 2^2\) o.e. \(= 20\) which is less than 29 | M1 | allow 2 for showing circle crosses \(x\) axis at \(-1\) and 9 or equiv for \(y\) (or showing one positive; one negative); 0 for graphical solutions |
| A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| showing midpt of \(AB = (4, 2)\) | 2 | in each method, two things need to be established. Allow M1 for the concept of what should be shown and A1 for correct completion with method shown |
| and showing \(AB = 2\sqrt{29}\) or showing \(AC\) or \(BC = \sqrt{29}\) or that A or B lie on circle | 2 | allow M1A0 for AB just shown as \(\sqrt{116}\) not \(2\sqrt{29}\); allow M1A0 for stating mid point of \(AB = (4,2)\) without working/method shown |
| or showing both A and B lie on circle (or \(AC = BC = \sqrt{29}\)), and showing \(AB = 2\sqrt{29}\) or that C is midpt of AB or that C is on AB or that gradients of AB and AC are the same or equiv. | 2 | NB showing \(AB = 2\sqrt{29}\) and C lies on AB is not sufficient – earns 2 marks only |
| or showing C is on AB and showing both A and B are on circle or \(AC = BC = \sqrt{29}\) | 2 | if M0, allow SC2 for accurate graph of circle drawn with compasses and AB joined with ruled line through C |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| grad \(AC\) or \(AB\) or \(BC = -\frac{5}{2}\) o.e. | M1 | may be seen in (iii) but only allow this M1 if they go on to use in this part |
| grad tgt \(= -1\)/their grad \(AC\) | M1 | allow for \(m_1 m_2 = -1\) used |
| tgt is \(y - 7 = \) their \(m(x - 2)\) o.e. | M1 | e.g. \(y = \) their \(mx + c\) then \((2, 7)\) subst; M0 if grad AC used |
| \(y = \frac{2}{5}x + \frac{31}{5}\) o.e. | A1 | condone \(y = 2/5x + c\) and \(c = 31/5\) o. |
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x-4)^2 - 16 + (y-2)^2 - 4 = 9$ o. | M2 | M1 for one completing square or for $(x-4)^2$ or $(y-2)^2$ expanded correctly, or starting with $(x-4)^2 + (y-2)^2 = r^2$: M1 for correct expn of at least one bracket and M1 for $9 + 20 = r^2$ o.e. |
| $\text{rad} = \sqrt{29}$ | B1 | or using $x^2 - 2gx + y^2 - 2fy + c = 0$: M1 for using centre is $(g, f)$ [must be quoted] and M1 for $r^2 = g^2 + f^2 - c$ |
**Part ii:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4^2 + 2^2$ o.e. $= 20$ which is less than 29 | M1 | allow 2 for showing circle crosses $x$ axis at $-1$ and 9 or equiv for $y$ (or showing one positive; one negative); 0 for graphical solutions |
| | A1 | |
**Part iii:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| showing midpt of $AB = (4, 2)$ | 2 | in each method, two things need to be established. Allow M1 for the concept of what should be shown and A1 for correct completion with method shown |
| and showing $AB = 2\sqrt{29}$ or showing $AC$ or $BC = \sqrt{29}$ or that A or B lie on circle | 2 | allow M1A0 for AB just shown as $\sqrt{116}$ not $2\sqrt{29}$; allow M1A0 for stating mid point of $AB = (4,2)$ without working/method shown |
| or showing both A and B lie on circle (or $AC = BC = \sqrt{29}$), and showing $AB = 2\sqrt{29}$ or that C is midpt of AB or that C is on AB or that gradients of AB and AC are the same or equiv. | 2 | NB showing $AB = 2\sqrt{29}$ and C lies on AB is not sufficient – earns 2 marks only |
| or showing C is on AB and showing both A and B are on circle or $AC = BC = \sqrt{29}$ | 2 | if M0, allow SC2 for accurate graph of circle drawn with compasses and AB joined with ruled line through C |
**Part iv:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| grad $AC$ or $AB$ or $BC = -\frac{5}{2}$ o.e. | M1 | may be seen in (iii) but only allow this M1 if they go on to use in this part |
| grad tgt $= -1$/their grad $AC$ | M1 | allow for $m_1 m_2 = -1$ used |
| tgt is $y - 7 = $ their $m(x - 2)$ o.e. | M1 | e.g. $y = $ their $mx + c$ then $(2, 7)$ subst; M0 if grad AC used |
| $y = \frac{2}{5}x + \frac{31}{5}$ o.e. | A1 | condone $y = 2/5x + c$ and $c = 31/5$ o. |
---2 A circle has equation $x ^ { 2 } + y ^ { 2 } - 8 x - 4 y = 9$.\\
(i) Show that the centre of this circle is $C ( 4,2 )$ and find the radius of the circle.\\
(ii) Show that the origin lies inside the circle.\\
(iii) Show that AB is a diameter of the circle, where A has coordinates ( 2,7 ) and B has coordinates $( 6 , - 3 )$.\\
(iv) Find the equation of the tangent to the circle at A . Give your answer in the form $y = m x + c$.
\hfill \mbox{\textit{OCR MEI C1 Q2 [13]}}