| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Tangency condition for line and curve |
| Difficulty | Standard +0.3 This is a standard tangency problem requiring substitution of a linear equation into a circle equation, followed by applying the discriminant condition (b²-4ac=0) for tangency. While it involves multiple steps and algebraic manipulation, it follows a well-established procedure taught in all A-level courses with no novel insight required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| Show x-coordinates of intersection of circle \(C: x^2 + y^2 - 6x - 14y = 40\) and line \(l: y = x + k\) satisfy \(2x^2 + (2k - 20)x + k^2 - 14k - 40 = 0\) | (2) | M1 Attempts to form equation with terms of form \(x^2\), \(x\), \(k^2\) and \(kx\) only using \(y = x + k\) and \(x^2 + y^2 - 6x - 14y = 40\) which must be appropriate form; A1 Uses correct and accurate algebra leading to given solution |
| Answer | Marks | Guidance |
|---|---|---|
| Find two values of \(k\) for which \(l\) is tangent to \(C\) | (4) | M1 Attempts to use \(b^2 - 4ac = 0\) with \(a = 2\), \(b = 2k - 20\), \(c = k^2 - 14k - 40\) and forms 3TQ equation in terms of \(k\); A1 Correct quadratic equation in \(k\); M1 Correct attempt to solve their 3TQ in \(k\); A1 \(k = 18\), \(k = -10\) |
**Part (a):**
Show x-coordinates of intersection of circle $C: x^2 + y^2 - 6x - 14y = 40$ and line $l: y = x + k$ satisfy $2x^2 + (2k - 20)x + k^2 - 14k - 40 = 0$ | (2) | M1 Attempts to form equation with terms of form $x^2$, $x$, $k^2$ and $kx$ only using $y = x + k$ and $x^2 + y^2 - 6x - 14y = 40$ which must be appropriate form; A1 Uses correct and accurate algebra leading to given solution
**Part (b):**
Find two values of $k$ for which $l$ is tangent to $C$ | (4) | M1 Attempts to use $b^2 - 4ac = 0$ with $a = 2$, $b = 2k - 20$, $c = k^2 - 14k - 40$ and forms 3TQ equation in terms of $k$; A1 Correct quadratic equation in $k$; M1 Correct attempt to solve their 3TQ in $k$; A1 $k = 18$, $k = -10$
---\begin{enumerate}
\item A circle $C$ has equation $x ^ { 2 } + y ^ { 2 } - 6 x - 14 y = 40$.
\end{enumerate}
The line $l$ has equation $y = x + k$, where $k$ is a constant.\\
a. Show that the $x$-coordinate of the points where $C$ and $l$ intersect are given by the solutions to the equation
$$2 x ^ { 2 } + ( 2 k - 20 ) x + k ^ { 2 } - 14 k - 40 = 0$$
b. Hence find the two values of $k$ for which $l$ is a tangent to $C$.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q14 [6]}}