| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Topic | Circles |
| Type | Tangent equation at a known point on circle |
| Difficulty | Standard +0.3 This is a two-part circle geometry question requiring standard techniques: (i) finding a tangent equation using the radius perpendicular property, and (ii) determining constraints on a parameter for a circle's position. Both parts are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| 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 |
\begin{enumerate}[label=(\roman*)]
\item A circle $C_1$ has equation
$$x^2 + y^2 + 18x - 2y + 30 = 0$$
The line $l$ is the tangent to $C_1$ at the point $P(-5, 7)$.
Find an equation of $l$ in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers to be found. [5]
\item A different circle $C_2$ has equation
$$x^2 + y^2 - 8x + 12y + k = 0$$
where $k$ is a constant.
Given that $C_2$ lies entirely in the 4th quadrant, find the range of possible values for $k$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q11 [9]}}