| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Tangent equation at a known point on circle |
| Difficulty | Moderate -0.3 Part (i) is a standard tangent-to-circle question requiring finding the center, computing the gradient of the radius, then using perpendicular gradients - routine AS-level technique. Part (ii) requires completing the square to find center/radius, then applying quadrant conditions, which is slightly more conceptual but still follows standard methods. Overall slightly easier than average due to straightforward application of learned techniques. |
| 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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^2+y^2+18x-2y+30=0 \Rightarrow (x+9)^2+(y-1)^2=\ldots\) | M1 | Implies centre \((\pm9, \pm1)\) |
| Centre \((-9, 1)\) | A1 | States or uses centre \((-9,1)\) |
| Gradient of line from \(P(-5,7)\) to \((-9,1)\): \(\frac{7-1}{-5+9} = \frac{3}{2}\) | M1 | Correct attempt to find gradient of radius using their \((-9,1)\) and \(P\) |
| Equation of tangent: \(y - 7 = -\frac{2}{3}(x+5)\) | dM1 | Complete strategy using perpendicular gradients; dependent on both previous M's |
| \(3y - 21 = -2x - 10 \Rightarrow 2x + 3y - 11 = 0\) | A1 | oe such as \(k(2x+3y-11)=0, k\in\mathbb{Z}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^2+y^2-8x+12y+k=0 \Rightarrow (x-4)^2+(y+6)^2 = 52-k\) | M1 | |
| Lies in Quadrant 4 if radius \(< 4 \Rightarrow 52-k < 4^2\) | M1 | |
| \(\Rightarrow k > 36\) | A1 | |
| \(52-k > 0 \Rightarrow\) Full solution \(36 < k < 52\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x \pm 4)^2 + (y \pm 6)^2 = P - k\), centre \((\mp 4, \mp 6)\), radius \(\sqrt{P-k}\) | M1 | Seen or implied by centre coordinates and radius |
| Radius \(< 4\) applied to obtain inequality in \(k\) only | M1 | Strategy that circle lies entirely within quadrant; condone \(\leq 4\) |
| \(k > 36\) | A1 | |
| \(36 < k < 52\) (from \(k > 36\) and \(52 - k > 0\)) | A1 | Rigorous argument; allow \(36 < k\ ``\ 52\) |
# Question 11:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2+y^2+18x-2y+30=0 \Rightarrow (x+9)^2+(y-1)^2=\ldots$ | M1 | Implies centre $(\pm9, \pm1)$ |
| Centre $(-9, 1)$ | A1 | States or uses centre $(-9,1)$ |
| Gradient of line from $P(-5,7)$ to $(-9,1)$: $\frac{7-1}{-5+9} = \frac{3}{2}$ | M1 | Correct attempt to find gradient of radius using their $(-9,1)$ and $P$ |
| Equation of tangent: $y - 7 = -\frac{2}{3}(x+5)$ | dM1 | Complete strategy using perpendicular gradients; dependent on both previous M's |
| $3y - 21 = -2x - 10 \Rightarrow 2x + 3y - 11 = 0$ | A1 | oe such as $k(2x+3y-11)=0, k\in\mathbb{Z}$ |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2+y^2-8x+12y+k=0 \Rightarrow (x-4)^2+(y+6)^2 = 52-k$ | M1 | |
| Lies in Quadrant 4 if radius $< 4 \Rightarrow 52-k < 4^2$ | M1 | |
| $\Rightarrow k > 36$ | A1 | |
| $52-k > 0 \Rightarrow$ Full solution $36 < k < 52$ | A1 | |
# Question 11(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x \pm 4)^2 + (y \pm 6)^2 = P - k$, centre $(\mp 4, \mp 6)$, radius $\sqrt{P-k}$ | M1 | Seen or implied by centre coordinates and radius |
| Radius $< 4$ applied to obtain inequality in $k$ only | M1 | Strategy that circle lies entirely within quadrant; condone $\leq 4$ |
| $k > 36$ | A1 | |
| $36 < k < 52$ (from $k > 36$ and $52 - k > 0$) | A1 | Rigorous argument; allow $36 < k\ ``\ 52$ |
---\begin{enumerate}
\item (i) A circle $C _ { 1 }$ has equation
\end{enumerate}
$$x ^ { 2 } + y ^ { 2 } + 18 x - 2 y + 30 = 0$$
The line $l$ is the tangent to $C _ { 1 }$ at the point $P ( - 5,7 )$.\\
Find an equation of $l$ in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers to be found.\\
(ii) A different circle $C _ { 2 }$ has equation
$$x ^ { 2 } + y ^ { 2 } - 8 x + 12 y + k = 0$$
where $k$ is a constant.\\
Given that $C _ { 2 }$ lies entirely in the 4th quadrant, find the range of possible values for $k$.
\hfill \mbox{\textit{Edexcel AS Paper 1 2020 Q11 [9]}}