| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Tangent equation at a known point on circle |
| Difficulty | Moderate -0.3 This is a structured multi-part question testing standard circle techniques (completing the square, distance calculations, perpendicular gradients for tangents). All parts are routine applications of formulae with clear guidance ('show that'), making it slightly easier than average despite requiring multiple steps. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.03f Circle properties: angles, chords, tangents |
| Answer | Marks | Guidance |
|---|---|---|
| i \((x - 4)^2 - 16 + (y - 2)^2 - 4 = 9\) o.e.; rad \(= \sqrt{29}\) | Marks: 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.) | Guidance: or using \(x^2 - 2gx + y^2 - 2fy + c = 0\); M1 for using centre is \((g, f)\) [must be quoted] and M1 for \(r^2 = g^2 + f^2 - c\) |
| ii \(4^2 + 2^2\) o.e.; \(= 20\) which is less than 29 | Marks: M1, A1 | Guidance: 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 (often using A and B from (iii) to draw circle) |
| iii showing midpt of AB \(= (4, 2)\); and showing \(AB = 2\sqrt{29}\) or showing \(AC\) or \(BC = \sqrt{29}\) or that A or B lie on circle | Marks: 2, 2 | Guidance: 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. |
| or showing C is on AB; and showing both A and B are on circle or \(AC = BC = \sqrt{29}\) | Marks: 2, 2 | Guidance: if M0, allow SC2 for accurate graph of circle drawn with compasses and AB joined with ruled line through C. |
| iv grad AC or AB or BC \(= -5/2\) o.e.; grad tgt \(= -1/\)their grad AC; tgt is \(y - 7 = \) their \(m(x - 2)\) o.e.; \(y = 2/5x + 31/5\) o.e. | Marks: M1, M1, M1, A1 | Guidance: allow for \(m \cdot m_\perp = -1\) used; eg \(y = \) their \(mx + c\) then \((2, 7)\) subst; M0 if grad AC used; condone \(y = 2/5x + c\) and \(c = 31/5\) o.e. |
**i** $(x - 4)^2 - 16 + (y - 2)^2 - 4 = 9$ o.e.; rad $= \sqrt{29}$ | **Marks:** 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.) | **Guidance:** or using $x^2 - 2gx + y^2 - 2fy + c = 0$; M1 for using centre is $(g, f)$ [must be quoted] and M1 for $r^2 = g^2 + f^2 - c$ | **Total:** 3
**ii** $4^2 + 2^2$ o.e.; $= 20$ which is less than 29 | **Marks:** M1, A1 | **Guidance:** 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 (often using A and B from (iii) to draw circle) | **Total:** 2
**iii** showing midpt of AB $= (4, 2)$; and showing $AB = 2\sqrt{29}$ or showing $AC$ or $BC = \sqrt{29}$ or that A or B lie on circle | **Marks:** 2, 2 | **Guidance:** 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. | **Total:** 4
or showing C is on AB; and showing both A and B are on circle or $AC = BC = \sqrt{29}$ | **Marks:** 2, 2 | **Guidance:** if M0, allow SC2 for accurate graph of circle drawn with compasses and AB joined with ruled line through C. | **Total:** 4
**iv** grad AC or AB or BC $= -5/2$ o.e.; grad tgt $= -1/$their grad AC; tgt is $y - 7 = $ their $m(x - 2)$ o.e.; $y = 2/5x + 31/5$ o.e. | **Marks:** M1, M1, M1, A1 | **Guidance:** allow for $m \cdot m_\perp = -1$ used; eg $y = $ their $mx + c$ then $(2, 7)$ subst; M0 if grad AC used; condone $y = 2/5x + c$ and $c = 31/5$ o.e. | **Total:** 4
**Total for Question 12:** 1312 A circle has equation $x ^ { 2 } + y ^ { 2 } - 8 x - 4 y = 9$.\\
(i) Show that the centre of this circle is $\mathrm { C } ( 4,2 )$ 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 2008 Q12 [13]}}