a) \(\displaystyle{\left({y}–{9}\right)}^{{2}}={8}{\left({x}-{2}\right)}\)

\(\displaystyle{\left({y}–{k}\right)}^{{2}}={4}{p}{\left({x}–{h}\right)}\)

\(\displaystyle{h}={2},{k}={9},\)

\(\displaystyle{4}{p}={8}\)

\(\displaystyle{p}=\frac{{8}}{{4}}={2}\)

vertex \(\displaystyle={\left({h},{k}\right)}={\left({2},{9}\right)}\)

focus \(\displaystyle={\left({4},{9}\right)}\)

directrix \(\displaystyle={x}={h}–{p}={2}–{2}={0}\)

b) \(\displaystyle{y}^{{2}}–{4}{y}={4}{x}–{2}^{{2}}\) – transform, as \(\displaystyle{a}^{{2}}+{b}^{{2}}+{2}{a}{b}={\left({a}+{b}\right)}^{{2}}\), so

\(\displaystyle{y}^{{2}}–{4}{y}+{4}={4}{x}+{4}+{4}\)

\(\displaystyle{\left({y}–{2}\right)}^{{2}}={4}{\left({x}+{2}\right)}\)

\(\displaystyle{h}=-{2},{k}={2}\)

\(\displaystyle{4}{a}={4}\)

\(\displaystyle{a}={1}\)

vertex \(\displaystyle={\left({h},{k}\right)}={\left(-{2},{2}\right)}\)

focus \(\displaystyle={\left(-{2},{3}\right)}\)

directx \(\displaystyle={h}–{p}=-{2}–{3}=-{5}\)

c) \(\displaystyle{\left({x}–{6}\right)}^{{2}}={4}{\left({y}–{2}\right)}\)

\(\displaystyle{\left({x}–{h}\right)}^{{2}}={4}{p}{\left({y}–{k}\right)}\)

\(\displaystyle{h}={6},{k}={2}\)

\(\displaystyle{4}{p}={4}\)

\(\displaystyle{p}={1}\)

vertex \(\displaystyle={\left({h},{k}\right)}={\left({6},{2}\right)}\)

focus \(\displaystyle={\left({h},{k}+{p}\right)}={\left({6},{3}\right)}\)

directrix \(\displaystyle={k}–{p}={2}–{1}={1}\)

\(\displaystyle{\left({y}–{k}\right)}^{{2}}={4}{p}{\left({x}–{h}\right)}\)

\(\displaystyle{h}={2},{k}={9},\)

\(\displaystyle{4}{p}={8}\)

\(\displaystyle{p}=\frac{{8}}{{4}}={2}\)

vertex \(\displaystyle={\left({h},{k}\right)}={\left({2},{9}\right)}\)

focus \(\displaystyle={\left({4},{9}\right)}\)

directrix \(\displaystyle={x}={h}–{p}={2}–{2}={0}\)

b) \(\displaystyle{y}^{{2}}–{4}{y}={4}{x}–{2}^{{2}}\) – transform, as \(\displaystyle{a}^{{2}}+{b}^{{2}}+{2}{a}{b}={\left({a}+{b}\right)}^{{2}}\), so

\(\displaystyle{y}^{{2}}–{4}{y}+{4}={4}{x}+{4}+{4}\)

\(\displaystyle{\left({y}–{2}\right)}^{{2}}={4}{\left({x}+{2}\right)}\)

\(\displaystyle{h}=-{2},{k}={2}\)

\(\displaystyle{4}{a}={4}\)

\(\displaystyle{a}={1}\)

vertex \(\displaystyle={\left({h},{k}\right)}={\left(-{2},{2}\right)}\)

focus \(\displaystyle={\left(-{2},{3}\right)}\)

directx \(\displaystyle={h}–{p}=-{2}–{3}=-{5}\)

c) \(\displaystyle{\left({x}–{6}\right)}^{{2}}={4}{\left({y}–{2}\right)}\)

\(\displaystyle{\left({x}–{h}\right)}^{{2}}={4}{p}{\left({y}–{k}\right)}\)

\(\displaystyle{h}={6},{k}={2}\)

\(\displaystyle{4}{p}={4}\)

\(\displaystyle{p}={1}\)

vertex \(\displaystyle={\left({h},{k}\right)}={\left({6},{2}\right)}\)

focus \(\displaystyle={\left({h},{k}+{p}\right)}={\left({6},{3}\right)}\)

directrix \(\displaystyle={k}–{p}={2}–{1}={1}\)