Step 1

a) Consider the equation \(\displaystyle{8}{x}^{{{2}}}\ -\ {2}{y}^{{{2}}}\ -\ {3}{x}\ +\ {2}{y}\ -\ {6}={0}\) It is known that the equation of conic section is in the form \(\displaystyle{A}{x}^{{{2}}}\ +\ {C}{y}^{{{2}}}\ +\ {D}{x}\ +\ {E}{y}\ +\ {F}={0}\) In the given equation, the coefficient of \(\displaystyle{x}^{{{2}}}\ \text{and}\ {y}^{{{2}}}\ \text{are}\ {A}={8}\ \text{and}\ {C}=\ -{2}\), respectively, which means \(\displaystyle{A}{C}=\ -{16},\) that is, \(\displaystyle{A}{C}\ {<}\ {0}\). Therefore, this is the equation of a hyperbola. Hence, the conic section of the given equation is a hyperbola.

Step 2

b) Consider the equation \(\displaystyle-{6}{y}^{{{2}}}\ +\ {4}{x}\ -\ {12}{y}\ -\ {24}={0}.\) It is known that the equation of conic section is in the form \(\displaystyle{A}{x}^{{{2}}}\ +\ {C}{y}^{{{2}}}\ +\ {D}{x}\ +\ {E}{y}\ +\ {F}={0}.\) Here, the given equation does not have the term of \(\displaystyle{x}^{{{2}}}\), which means \(\displaystyle{A}={0},\) that is, \(\displaystyle{A}{C}={0}\). Therefore, this is the equation of parabola. Hence, the conic section of the given equation is a parabola. Step 3 c) Consider the equation \(\displaystyle-{9}{x}^{{{2}}}\ -\ {4}{y}^{{{2}}}\ -\ {18}{x}\ +\ {12}{y}={0}.\) It is known that the equation of conic section is in the form \(\displaystyle{A}{x}^{{{2}}}\ +\ {C}{y}^{{{2}}}\ +\ {D}{x}\ +\ {E}{y}\ +\ {F}={0}\). In the given equation, the coefficient of \(\displaystyle{x}^{{{2}}}\) and \(\displaystyle{y}^{{{2}}}\) are \(\displaystyle{A}=\ -{9}\) and \(\displaystyle{C}=\ -{4},\) respectively, which means \(\displaystyle{A}{C}={36},\) that is, \(\displaystyle{A}{C}\ {>}\ {0}\). Therefore, this is the equation of an ellipse. Hence, the conic section of the given equation is an ellipse.