G1.3 Given that $c$ is a positive number. If there is only one straight line which passes through point $\mathrm{A}(1, c)$ and meets the curve $C: x^{2}+y^{2}-2 x-2 y-7=0$ at only one point, find the value of $c$.
The curve is a circle.
There is only one straight line which passes through point $\mathrm{A}$ and meets the curve at only one point $\Rightarrow$ the straight line is a tangent and the point $A(1, c)$ lies on the circle.
(otherwise two tangents can be drawn if $A$ lies outside the circle)
Put $x=1, y=c$ into the circle.
$$
\begin{array}{l}
1+c^{2}-2-2 c-7=0 \\
c^{2}-2 c-8=0 \\
(c-4)(c+2)=0 \\
c=4 \text { or } c=-2 \text { (rejected) }
\end{array}
$$