Concept used:
Write the general equation of the hyperbola.
\(\displaystyle{\frac{{{\left({x}-{h}\right)}^{{{2}}}}}{{{a}^{{{2}}}}}}-{\frac{{{\left({y}-{k}\right)}^{{{2}}}}}{{{b}^{{{2}}}}}}={1}\)
The centre of the hyperbola is (h,k), the vertices are \(\displaystyle{\left({h}\pm{a},{k}\right)}\), and foci are \(\displaystyle{\left({h}\pm{c},{k}\right)}\)
Write the expression for c.
\(\displaystyle{c}=\sqrt{{{a}^{{{2}}}+{b}^{{{2}}}}}\)
The given equation is,
\(\displaystyle{\left({\frac{{{x}-{4}}}{{{5}}}}\right)}^{{{2}}}-{\left({\frac{{{y}+{3}}}{{{6}}}}\right)}^{{{2}}}={1}\)

\(\displaystyle{\frac{{{\left({x}-{4}\right)}^{{{2}}}}}{{{5}^{{{2}}}}}}-{\frac{{{\left({y}+{3}\right)}^{{{2}}}}}{{{6}^{{{2}}}}}}={1}\) Write the general equation of the hyperbola. \(\displaystyle{\frac{{{\left({x}-{h}\right)}^{{{2}}}}}{{{a}^{{{2}}}}}}-{\frac{{{\left({y}-{k}\right)}^{{{2}}}}}{{{b}}}}={1}\) Compare the given equation with the general equation of the hyperbola to get, \(\displaystyle{h}={4}\)

\(\displaystyle{k}=-{3}\)

\(\displaystyle{a}={5}\)

\(\displaystyle{b}={6}\) Write the expression for c. \(\displaystyle{c}=\sqrt{{{a}^{{{2}}}+{b}^{{{2}}}}}\) Substitute 6 for a and 5 for b. \(\displaystyle{c}=\sqrt{{{5}^{{{2}}}+{6}^{{{2}}}}}\)

\(\displaystyle=\sqrt{{{25}+{36}}}\)

\(\displaystyle=\sqrt{{{61}}}\) The vertices of the conic section are, \(\displaystyle{\left({h}\pm{a},{k}\right)}={\left({4}\pm{5},-{3}\right)}\)

\(\displaystyle={\left({4}+{5},-{3}\right)}{\left({4}-{5},-{3}\right)}\)

\(\displaystyle={\left({9},-{3}\right)}{\left(-{1},-{3}\right)}\) The foci of the conic section are, \(\displaystyle{\left({h}\pm{a},{k}\right)}={\left({4}\pm\sqrt{{{61}}},-{3}\right)}\)

\(\displaystyle={\left({4}+\sqrt{{{61}}},-{3}\right)}{\left({4}-\sqrt{{{61}}},-{3}\right)}\) Therefore, the vertices and foci of the conic section are \(\displaystyle{\left({9},-{3}\right)}{\left(-{1},-{3}\right)}\) and PSK(4+\sqrt{61},-3)(4-\sqrt{61},-3) respectively.

\(\displaystyle{\frac{{{\left({x}-{4}\right)}^{{{2}}}}}{{{5}^{{{2}}}}}}-{\frac{{{\left({y}+{3}\right)}^{{{2}}}}}{{{6}^{{{2}}}}}}={1}\) Write the general equation of the hyperbola. \(\displaystyle{\frac{{{\left({x}-{h}\right)}^{{{2}}}}}{{{a}^{{{2}}}}}}-{\frac{{{\left({y}-{k}\right)}^{{{2}}}}}{{{b}}}}={1}\) Compare the given equation with the general equation of the hyperbola to get, \(\displaystyle{h}={4}\)

\(\displaystyle{k}=-{3}\)

\(\displaystyle{a}={5}\)

\(\displaystyle{b}={6}\) Write the expression for c. \(\displaystyle{c}=\sqrt{{{a}^{{{2}}}+{b}^{{{2}}}}}\) Substitute 6 for a and 5 for b. \(\displaystyle{c}=\sqrt{{{5}^{{{2}}}+{6}^{{{2}}}}}\)

\(\displaystyle=\sqrt{{{25}+{36}}}\)

\(\displaystyle=\sqrt{{{61}}}\) The vertices of the conic section are, \(\displaystyle{\left({h}\pm{a},{k}\right)}={\left({4}\pm{5},-{3}\right)}\)

\(\displaystyle={\left({4}+{5},-{3}\right)}{\left({4}-{5},-{3}\right)}\)

\(\displaystyle={\left({9},-{3}\right)}{\left(-{1},-{3}\right)}\) The foci of the conic section are, \(\displaystyle{\left({h}\pm{a},{k}\right)}={\left({4}\pm\sqrt{{{61}}},-{3}\right)}\)

\(\displaystyle={\left({4}+\sqrt{{{61}}},-{3}\right)}{\left({4}-\sqrt{{{61}}},-{3}\right)}\) Therefore, the vertices and foci of the conic section are \(\displaystyle{\left({9},-{3}\right)}{\left(-{1},-{3}\right)}\) and PSK(4+\sqrt{61},-3)(4-\sqrt{61},-3) respectively.