Concept Used: The general equation of conic \(\displaystyle{a}{x}^{2}\ +\ {2}{h}{x}{y}\ +\ {b}{y}^{2}\ +\ {2}{g}{x}\ +\ {2}{f}{y}\ +\ {c}={0}\) has the discriminant \(\displaystyle\Delta\) and its given as \(\displaystyle\Delta=\mid{a}{h}{g}{h}{b}{f}{g}{f}{c}\mid={a}{b}{c}\ +\ {2}{f}{g}{h}\ -\ {a}{f}^{2}\ -\ {b}{g}^{2}\ -\ {c}{h}^{2}\) The general equation of conic \(\displaystyle{a}{x}^{2}\ +\ {2}{h}{x}{y}\ +\ {b}{y}^{2}\ +\ {2}{g}{x}\ +\ {2}{f}{y}\ +\ {c}={0}\) can also be written in the following way \(\displaystyle{A}{x}^{2}\ +\ {B}{X}{Y}\ +\ {C}{y}^{2}\ +\ {D}{x}\ +\ {E}{y}\ +\ {F}={0}\) If \(\displaystyle\Delta\ne{q}{0},\) then the equation is nondegenerate and under this case \(\displaystyle var \neg{{h}}\in{g}\ \text{If}\ {B}^{2}\ -\ {4}{A}{C}\ {>}\ {0},\) it represents a hyperbola and a rectangular hyperbola if \(\displaystyle{\left({A}\ +\ {C}={0}\right)}\)

\(\displaystyle{v}{a}{r}\neg{h}\in{g}\ \text{If}\ {B}^{2}\ -\ {4}{A}{C}={0},\) it represents a parabola. \(\displaystyle{v}{a}{r}\neg{h}\in{g}\ \text{If}\ {B}^{2}\ -\ {4}{A}{C}\ {<}\ {0},\text{it represents a circle }{\left({A}={C},\ {B}={0}\right)}\) or ellipse \(\displaystyle{A}\ne{q}{C}\) Calculations: The given equation is: \(\displaystyle{x}^{2}\ -\ {2}{x}{y}\ +\ {y}^{2}\ +\ {24}{x}\ -\ {8}={0}\) and the equation is nondegenerate, hence \(\displaystyle\Delta\ne{q}{0}\). Now let's find the value of \(\displaystyle{B}^{2}\ -\ {4}{A}{C}\). From the given equation we have \(\displaystyle{A}={1},\ {B}=\ -{2},\ {C}={1},\) so on substitution we get \(\displaystyle\Rightarrow\ {B}^{2}\ -\ {4}{A}{C}={\left(-{2}\right)}^{2}\ -\ {4}\ \cdot\ {1}\ \cdot\ {1}\ \Rightarrow\ {B}^{2}\ -\ {4}{A}{C}={4}\ -\ {4}\ \Rightarrow\ {B}^{2}\ -\ {4}{A}{C}={0}\) Hence the given equation represents a Parabola. The plot of the given curve is as below: Conclusion: The given equation represents a Parabola and the plot is shown above.