| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Find equations of tangent lines with given gradient or from external point using discriminant |
| Difficulty | Standard +0.3 This is a standard circle question requiring completing the square to find centre and radius, then using perpendicular distance from centre to tangent equals radius. Part (a) is routine manipulation, while part (b) involves applying the tangent condition but follows a well-practiced method with no novel insight required. Slightly easier than average due to being a textbook-style multi-part question. |
| Spec | 1.03b Straight lines: parallel and perpendicular relationships1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| i. \(C = (-1, 3)\) | M1, A1 | Attempts to complete the square; Gives the centre correctly |
| ii. \(r = 5\sqrt{2}\) | A1 | Correct answer only |
| Answer | Marks | Guidance |
|---|---|---|
| \(l_1: y = -7x - 54\), \(l_2: y = -7x + 46\) | M1, A1, M1, A1, M1, A1, M1, A1 | Attempts to write an equation of radius/diameter of circle; Correct equation; Substitutes their equation of radius into equation of the circle and obtain a quadratic equation in terms of \(x\); Factorise their quadratic equation and reaching a value of \(x\); Both \(x\) values are correct; Substituting \(x\) values into equation of radius and obtaining \(y\) values correctly; Attempts to write an equation of the tangents with their \(x\) and \(y\) values; Both equation of tangents are correct |
### Part a:
**i.** $C = (-1, 3)$ | M1, A1 | Attempts to complete the square; Gives the centre correctly
**ii.** $r = 5\sqrt{2}$ | A1 | Correct answer only
### Part b:
$l_1: y = -7x - 54$, $l_2: y = -7x + 46$ | M1, A1, M1, A1, M1, A1, M1, A1 | Attempts to write an equation of radius/diameter of circle; Correct equation; Substitutes their equation of radius into equation of the circle and obtain a quadratic equation in terms of $x$; Factorise their quadratic equation and reaching a value of $x$; Both $x$ values are correct; Substituting $x$ values into equation of radius and obtaining $y$ values correctly; Attempts to write an equation of the tangents with their $x$ and $y$ values; Both equation of tangents are correctA curve with centre $C$ has equation
$$x^2 + y^2 + 2x - 6y - 40 = 0$$
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item State the coordinates of $C$.
\item Find the radius of the circle, giving your answer as $r = n\sqrt{2}$. [3]
\end{enumerate}
\item The line $l$ is a tangent to the circle and has gradient $-7$. Find two possible equations for $l$, giving your answers in the form $y = mx + c$. [8]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 Q14 [11]}}