| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2022 |
| Session | October |
| Marks | 7 |
| Topic | Circles |
| Type | Find parameter values for tangency using discriminant |
| Difficulty | Standard +0.3 This is a straightforward coordinate geometry problem requiring standard techniques: finding the circle equation (center at (r,r) since it touches both axes), substituting the line equation, and using the discriminant condition for tangency. The algebra is routine and the setup is clearly guided, making it slightly easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
A circle $C$ with radius $r$
\begin{itemize}
\item lies only in the 1st quadrant
\item touches the $x$-axis and touches the $y$-axis
\end{itemize}
The line $l$ has equation $2x + y = 12$
\begin{enumerate}[label=(\alph*)]
\item Show that the $x$ coordinates of the points of intersection of $l$ with $C$ satisfy
$$5x^2 + (2r - 48)x + (r^2 - 24r + 144) = 0$$ [3]
\end{enumerate}
Given also that $l$ is a tangent to $C$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the two possible values of $r$, giving your answers as fully simplified surds. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2022 Q10 [7]}}