$\begin{aligned}
& \text { (a) } x^2+y^2+2 x-4 y+1=0 \\
& \therefore g_1=1, f_1=-2, c_1=1 \\
& x^2+y^2-4 x-2 y+c=0 \\
& \therefore g_1=-2, f_2=-1, c_2=\mathrm{c}
\end{aligned}$
Now, angle between the circles is
$\begin{aligned}
& \cos (\theta)=\frac{c_1+c_2-2\left(\mathrm{~g}_1 \mathrm{~g}_2+\mathrm{f}_1 \mathrm{f}_2\right)}{2 \sqrt{g_1^2+f_1^2-c_1} \sqrt{g_2^2+f_2^2-c_2}} \\
& \Rightarrow \cos \left(\frac{\pi}{4}\right)=\frac{1+c-2(1 \times(-2)+(-2)(-1))}{\sqrt[2]{1+4-1} \sqrt{4+1-c}} \\
& \Rightarrow \frac{1}{\sqrt{2}}=\frac{1+c}{4 \sqrt{5-c}} \Rightarrow 16(5-c)=2(1+c)^2 \\
& \Rightarrow c^2+10 c-39=0 \Rightarrow(c-3)(c+13)=0 \\
& \Rightarrow c=3 \text { or } c=-13
\end{aligned}$