<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}


{\bf Question}

\begin{description}
\item[(a)]Find the equation of a sphere centre ${\bf c}$ and
radius $a$.
\item[(b)] Show that the equation of the tangent plane at a point
${\bf d}$ on the sphere is $${\bf r} \cdot {\bf d} - {\bf c} \cdot
({\bf r} + {\bf d}) + k =0$$ where $k$ is some scalar to be
determined.
\end{description}

\vspace{.25in}

{\bf Answer}

\begin{description}
\item[(a)] $|r-c|^2 = a^2$ or $({\bf r -c}) \cdot ({\bf r -a}) =
a^2 \hspace{.2in} {\bf r \cdot r} - 2{\bf r \cdot c} = a^2 - c^2$
\item[(b)] a normal vector to the plane i s${\bf d-c}$ so the
equation is ${\bf r \cdot (d-c)} = k$ Thus the equation is ${\bf
r(d-c) = d \cdot (d-c)}$ $${\bf r \cdot d - c(r+d) - d \cdot d } =
0$$
\end{description}

\end{document}
</pre></body></html>