| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2020 |
| Session | February |
| Marks | 9 |
| Topic | Circles |
| Type | Two circles intersection or tangency |
| Difficulty | Standard +0.3 This is a straightforward circles question requiring completion of the square to find centres/radii, verification that x=0 is tangent (routine check that distance from centre equals radius), finding a line through two points, and using the property that common tangents have equal perpendicular distances from both centres. All techniques are standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.03a 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 circle1.03f Circle properties: angles, chords, tangents |
11 Two circles have equations\\
$x ^ { 2 } + y ^ { 2 } - 4 x = 0 \quad$ and\\
$x ^ { 2 } + y ^ { 2 } - 6 x - 12 y + 36 = 0$.
\begin{enumerate}[label=(\alph*)]
\item Find the centre and radius of each circle and hence show that the $y$-axis is a tangent to both circles.
\item Find the equation of the line through the centres of both circles.
\item Determine the gradient of a line other than the $y$-axis which is a tangent to both circles.\\
\section*{ADDITIONAL ANSWER SPACE}
If additional space is required, you should use the following lined page(s). The question number(s) must be clearly shown. nunber(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2020 Q11 [9]}}