| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Session | Specimen |
| Marks | 8 |
| 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 two-part question requiring standard techniques: (a) finding circle equation from centre and a point using distance formula, (b) using perpendicularity of radius and tangent to find the tangent line equation. Both parts are routine AS-level procedures with no problem-solving insight required, making it easier than average but not trivial. |
| 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, tangents1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Attempts \(\sqrt{(2--2)^2+(5-3)^2}\) or radius\(^2\) | M1 | Attempts to find the radius |
| Attempts \((x-2)^2+(y-5)^2 = {r}^2\) | M1 | States or uses correct circle form with centre \((2,5)\) |
| \((x-2)^2+(y-5)^2 = 20\) | A1 | Correct equation. Alt: \(x^2+y^2-4x-10y+9=0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Gradient of \(OP\): \(\frac{5-3}{2--2}=\frac{1}{2}\) | M1 | Attempts gradient of \(OP\) where \(O\) is centre of \(C\) |
| Equation of \(l\): \(-2=\frac{y-3}{x+2}\) | dM1 | Complete strategy for equation of \(l\) using perpendicular gradient to \(OP\) and point \((-2,3)\) |
| \(y=-2x-1\) | A1 | Any correct form |
| Substitute \(x=2\) into \(y=-2x-1\) | M1 | Key step: substitutes \((2,k)\) into their \(y=-2x-1\) |
| \(k=-5\) | A1 | Correct value |
## Question 10:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| Attempts $\sqrt{(2--2)^2+(5-3)^2}$ or radius$^2$ | M1 | Attempts to find the radius |
| Attempts $(x-2)^2+(y-5)^2 = {r}^2$ | M1 | States or uses correct circle form with centre $(2,5)$ |
| $(x-2)^2+(y-5)^2 = 20$ | A1 | Correct equation. Alt: $x^2+y^2-4x-10y+9=0$ |
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| Gradient of $OP$: $\frac{5-3}{2--2}=\frac{1}{2}$ | M1 | Attempts gradient of $OP$ where $O$ is centre of $C$ |
| Equation of $l$: $-2=\frac{y-3}{x+2}$ | dM1 | Complete strategy for equation of $l$ using perpendicular gradient to $OP$ and point $(-2,3)$ |
| $y=-2x-1$ | A1 | Any correct form |
| Substitute $x=2$ into $y=-2x-1$ | M1 | Key step: substitutes $(2,k)$ into their $y=-2x-1$ |
| $k=-5$ | A1 | Correct value |
---\begin{enumerate}
\item A circle $C$ has centre $( 2,5 )$. Given that the point $P ( - 2,3 )$ lies on $C$.\\
(a) find an equation for $C$.
\end{enumerate}
The line $l$ is the tangent to $C$ at the point $P$. The point $Q ( 2 , k )$ lies on $l$.\\
(b) Find the value of $k$.
\hfill \mbox{\textit{Edexcel AS Paper 1 Q10 [8]}}